Module 1 Introduction to Ordinary Differential Equations Mr
![Module 1 Introduction to Ordinary Differential Equations Mr Peter Bier Module 1 Introduction to Ordinary Differential Equations Mr Peter Bier](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-1.jpg)
![Ordinary Differential Equations n n n Slide number 2 Where do ODEs arise? Notation Ordinary Differential Equations n n n Slide number 2 Where do ODEs arise? Notation](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-2.jpg)
![Where do ODE’s arise n n All branches of Engineering Economics Biology and Medicine Where do ODE’s arise n n All branches of Engineering Economics Biology and Medicine](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-3.jpg)
![Example – Newton’s Law of Cooling n This is a model of how the Example – Newton’s Law of Cooling n This is a model of how the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-4.jpg)
![Example – Swinging of a pendulum Newton’s 2 nd law for a rotating object: Example – Swinging of a pendulum Newton’s 2 nd law for a rotating object:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-5.jpg)
![Notation and Definitions n n Slide number 6 Order Linearity Homogeneity Initial Value/Boundary value Notation and Definitions n n Slide number 6 Order Linearity Homogeneity Initial Value/Boundary value](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-6.jpg)
![Order n The order of a differential equation is just the order of highest Order n The order of a differential equation is just the order of highest](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-7.jpg)
![Linearity n n Slide number 8 The important issue is how the unknown y Linearity n n Slide number 8 The important issue is how the unknown y](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-8.jpg)
![Linearity - Examples is linear is non-linear Slide number 9 Linearity - Examples is linear is non-linear Slide number 9](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-9.jpg)
![Linearity – Summary Linear Non-linear or Slide number 10 Linearity – Summary Linear Non-linear or Slide number 10](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-10.jpg)
![Linearity – Special Property If a linear homogeneous ODE has solutions: and then: where Linearity – Special Property If a linear homogeneous ODE has solutions: and then: where](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-11.jpg)
![Linearity – Special Property Example: has solutions and Check Therefore Check Slide number 12 Linearity – Special Property Example: has solutions and Check Therefore Check Slide number 12](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-12.jpg)
![Life is mostly linear!!! § Most ODEs that arise in engineering are linear with Life is mostly linear!!! § Most ODEs that arise in engineering are linear with](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-13.jpg)
![Approximately Linear – Swinging pendulum example n The accurate non-linear equation for a swinging Approximately Linear – Swinging pendulum example n The accurate non-linear equation for a swinging](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-14.jpg)
![Homogeniety n n n Slide number 15 Put all the terms of the equation Homogeniety n n n Slide number 15 Put all the terms of the equation](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-15.jpg)
![Initial Value/Boundary value problems n Problems that involve time are represented by an ODE Initial Value/Boundary value problems n Problems that involve time are represented by an ODE](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-16.jpg)
![Example n n n n Slide number 17 1 st order Linear Nonhomogeneous Initial Example n n n n Slide number 17 1 st order Linear Nonhomogeneous Initial](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-17.jpg)
![Example n n n n Slide number 18 2 nd order Nonlinear Homogeneous Initial Example n n n n Slide number 18 2 nd order Nonlinear Homogeneous Initial](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-18.jpg)
![Solution Methods - Direct Integration n n Slide number 19 This method works for Solution Methods - Direct Integration n n Slide number 19 This method works for](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-19.jpg)
![Direct Integration n n Slide number 20 y is called the unknown or dependent Direct Integration n n Slide number 20 y is called the unknown or dependent](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-20.jpg)
![Direct Integration – Example Find the velocity of a car that is accelerating from Direct Integration – Example Find the velocity of a car that is accelerating from](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-21.jpg)
![Bending of a beam - Example A beam under uniform load Beam theory gives Bending of a beam - Example A beam under uniform load Beam theory gives](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-22.jpg)
![Bending of a beam - Solution § Step 1: Integrate § Step 2: Integrate Bending of a beam - Solution § Step 1: Integrate § Step 2: Integrate](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-23.jpg)
![Bending of a beam - Solution § Step 3: Use the boundary conditions to Bending of a beam - Solution § Step 3: Use the boundary conditions to](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-24.jpg)
![Solution Methods - Separation The separation method applies only to 1 st order ODEs. Solution Methods - Separation The separation method applies only to 1 st order ODEs.](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-25.jpg)
![Separation – General Idea First Separate: Then integrate LHS with respect to y, RHS Separation – General Idea First Separate: Then integrate LHS with respect to y, RHS](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-26.jpg)
![Separation - Example Separate: Now integrate: Slide number 27 Separation - Example Separate: Now integrate: Slide number 27](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-27.jpg)
![Cooling of a cup of coffee Amount of heat in a cup of coffee: Cooling of a cup of coffee Amount of heat in a cup of coffee:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-28.jpg)
![Cooling of a cup of coffee Newton’s law of cooling: n Heat lost to Cooling of a cup of coffee Newton’s law of cooling: n Heat lost to](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-29.jpg)
![Cooling of a cup of coffee Substitute in rearrange where Now we solve the Cooling of a cup of coffee Substitute in rearrange where Now we solve the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-30.jpg)
![Cooling of a cup of coffee Solution § Step 1: Separate § Step 2: Cooling of a cup of coffee Solution § Step 1: Separate § Step 2:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-31.jpg)
![Cooling of a cup of coffee Solution § Step 3: Use Initial Condition § Cooling of a cup of coffee Solution § Step 3: Use Initial Condition §](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-32.jpg)
![Solution Methods - Integrating Factor The integrating factor method is used for nonhomogeneous linear Solution Methods - Integrating Factor The integrating factor method is used for nonhomogeneous linear](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-33.jpg)
![Integrating Factor – Example There are several ways to solve this problem but we Integrating Factor – Example There are several ways to solve this problem but we](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-34.jpg)
![Integrating Factor – Example Product Rule: The basic idea is that if we multiply Integrating Factor – Example Product Rule: The basic idea is that if we multiply](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-35.jpg)
![Integrating Factor – Example We will look ahead, use the answer and show it Integrating Factor – Example We will look ahead, use the answer and show it](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-36.jpg)
![Integrating Factor – Example We can rewrite the equation as: or Now we can Integrating Factor – Example We can rewrite the equation as: or Now we can](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-37.jpg)
![Integrating Factor – Example Rearrange to make this explicit in y Now use the Integrating Factor – Example Rearrange to make this explicit in y Now use the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-38.jpg)
![How do we calculate the integrating factor? n n Let us now pretend we How do we calculate the integrating factor? n n Let us now pretend we](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-39.jpg)
![How do we calculate the integrating factor? n Then the ODE becomes § Now How do we calculate the integrating factor? n Then the ODE becomes § Now](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-40.jpg)
![How do we calculate the integrating factor? n Now we can integrate once we How do we calculate the integrating factor? n Now we can integrate once we](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-41.jpg)
![Finding the integrating factor in general n Given the general form of a nonhomogeneous Finding the integrating factor in general n Given the general form of a nonhomogeneous](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-42.jpg)
![Finding the integrating factor in general n Step 1: Multiply by Φ: n Step Finding the integrating factor in general n Step 1: Multiply by Φ: n Step](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-43.jpg)
![Finding the integrating factor in general n Step 4: Combine terms on the LHS: Finding the integrating factor in general n Step 4: Combine terms on the LHS:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-44.jpg)
![Finding the integrating factor in general Notes: n After you have been through the Finding the integrating factor in general Notes: n After you have been through the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-45.jpg)
![Solving an example using the integrating factor method Step 1: Put the ODE into Solving an example using the integrating factor method Step 1: Put the ODE into](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-46.jpg)
![Solving an example using the integrating factor method Step 3: Multiply by the integrating Solving an example using the integrating factor method Step 3: Multiply by the integrating](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-47.jpg)
![Solving an example using the integrating factor method Step 6: Use the initial conditions Solving an example using the integrating factor method Step 6: Use the initial conditions](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-48.jpg)
![Exponential substitution n n The exponential trial or guessing method can be used for Exponential substitution n n The exponential trial or guessing method can be used for](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-49.jpg)
![Characteristic Equations n gives the differentials n An algebraic characteristic equation comes from substituting Characteristic Equations n gives the differentials n An algebraic characteristic equation comes from substituting](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-50.jpg)
![Exponential Trial - Example Try n Cancelling out equation n Substituting this back into Exponential Trial - Example Try n Cancelling out equation n Substituting this back into](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-51.jpg)
![Solving Guide General Form Slide number 52 Description Solving Method § § 1 st Solving Guide General Form Slide number 52 Description Solving Method § § 1 st](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-52.jpg)
![Solving Guide General Form Slide number 53 Description Solving Method § § 2 nd Solving Guide General Form Slide number 53 Description Solving Method § § 2 nd](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-53.jpg)
![Solving Guide General Form Slide number 54 Description Solving Method § § 2 nd Solving Guide General Form Slide number 54 Description Solving Method § § 2 nd](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-54.jpg)
- Slides: 54
![Module 1 Introduction to Ordinary Differential Equations Mr Peter Bier Module 1 Introduction to Ordinary Differential Equations Mr Peter Bier](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-1.jpg)
Module 1 Introduction to Ordinary Differential Equations Mr Peter Bier
![Ordinary Differential Equations n n n Slide number 2 Where do ODEs arise Notation Ordinary Differential Equations n n n Slide number 2 Where do ODEs arise? Notation](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-2.jpg)
Ordinary Differential Equations n n n Slide number 2 Where do ODEs arise? Notation and Definitions Solution methods for 1 st order ODEs
![Where do ODEs arise n n All branches of Engineering Economics Biology and Medicine Where do ODE’s arise n n All branches of Engineering Economics Biology and Medicine](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-3.jpg)
Where do ODE’s arise n n All branches of Engineering Economics Biology and Medicine Chemistry, Physics etc Anytime you wish to find out how something changes with time (and sometimes space) Slide number 3
![Example Newtons Law of Cooling n This is a model of how the Example – Newton’s Law of Cooling n This is a model of how the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-4.jpg)
Example – Newton’s Law of Cooling n This is a model of how the temperature of an object changes as it loses heat to the surrounding atmosphere: Temperature of the object: Room Temperature: Newton’s laws states: “The rate of change in the temperature of an object is proportional to the difference in temperature between the object and the room temperature” Form ODE Solve ODE Where Slide number 4 is the initial temperature of the object.
![Example Swinging of a pendulum Newtons 2 nd law for a rotating object Example – Swinging of a pendulum Newton’s 2 nd law for a rotating object:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-5.jpg)
Example – Swinging of a pendulum Newton’s 2 nd law for a rotating object: n Moment of inertia x angular acceleration = Net external torque q rearrange and divide through by ml 2 l where mg Slide number 5 This equation is very difficult to solve.
![Notation and Definitions n n Slide number 6 Order Linearity Homogeneity Initial ValueBoundary value Notation and Definitions n n Slide number 6 Order Linearity Homogeneity Initial Value/Boundary value](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-6.jpg)
Notation and Definitions n n Slide number 6 Order Linearity Homogeneity Initial Value/Boundary value problems
![Order n The order of a differential equation is just the order of highest Order n The order of a differential equation is just the order of highest](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-7.jpg)
Order n The order of a differential equation is just the order of highest derivative used. 2 nd order . 3 rd order Slide number 7
![Linearity n n Slide number 8 The important issue is how the unknown y Linearity n n Slide number 8 The important issue is how the unknown y](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-8.jpg)
Linearity n n Slide number 8 The important issue is how the unknown y appears in the equation. A linear equation involves the dependent variable (y) and its derivatives by themselves. There must be no "unusual" nonlinear functions of y or its derivatives. A linear equation must have constant coefficients, or coefficients which depend on the independent variable (t). If y or its derivatives appear in the coefficient the equation is non-linear.
![Linearity Examples is linear is nonlinear Slide number 9 Linearity - Examples is linear is non-linear Slide number 9](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-9.jpg)
Linearity - Examples is linear is non-linear Slide number 9
![Linearity Summary Linear Nonlinear or Slide number 10 Linearity – Summary Linear Non-linear or Slide number 10](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-10.jpg)
Linearity – Summary Linear Non-linear or Slide number 10
![Linearity Special Property If a linear homogeneous ODE has solutions and then where Linearity – Special Property If a linear homogeneous ODE has solutions: and then: where](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-11.jpg)
Linearity – Special Property If a linear homogeneous ODE has solutions: and then: where a and b are constants, is also a solution. Slide number 11
![Linearity Special Property Example has solutions and Check Therefore Check Slide number 12 Linearity – Special Property Example: has solutions and Check Therefore Check Slide number 12](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-12.jpg)
Linearity – Special Property Example: has solutions and Check Therefore Check Slide number 12 is also a solution:
![Life is mostly linear Most ODEs that arise in engineering are linear with Life is mostly linear!!! § Most ODEs that arise in engineering are linear with](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-13.jpg)
Life is mostly linear!!! § Most ODEs that arise in engineering are linear with constant coefficients. § In many cases they are approximate versions of more complex nonlinear models but they are sufficiently accurate for most purposes. Often they work OK for small amplitude disturbances but for large amplitude behaviour nonlinearities start to have some effect. § For linear systems the independent of amplitude. § The coefficients in the ODE correspond to system parameters and are usually constant. § Sometimes nonlinearities are important and there have been some important failures because nonlinearities were not understood e. g. the collapse of the Tacoma Narrows bridge. Slide number 13 qualitative behaviour is
![Approximately Linear Swinging pendulum example n The accurate nonlinear equation for a swinging Approximately Linear – Swinging pendulum example n The accurate non-linear equation for a swinging](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-14.jpg)
Approximately Linear – Swinging pendulum example n The accurate non-linear equation for a swinging pendulum is: n But for small angles of swing this can be approximated by the linear ODE: Slide number 14
![Homogeniety n n n Slide number 15 Put all the terms of the equation Homogeniety n n n Slide number 15 Put all the terms of the equation](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-15.jpg)
Homogeniety n n n Slide number 15 Put all the terms of the equation which involve the dependent variable on the LHS. Homogeneous: If there is nothing left on the RHS the equation is homogeneous (unforced or free) Nonhomogeneous: If there are terms involving t (or constants) - but not y - left on the RHS the equation is nonhomogeneous (forced)
![Initial ValueBoundary value problems n Problems that involve time are represented by an ODE Initial Value/Boundary value problems n Problems that involve time are represented by an ODE](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-16.jpg)
Initial Value/Boundary value problems n Problems that involve time are represented by an ODE together with initial values. n Problems that involve space (just one dimension) are also governed by an ODE but what is happening at the ends of the region of interest has to be specified as well by boundary conditions. Slide number 16
![Example n n n n Slide number 17 1 st order Linear Nonhomogeneous Initial Example n n n n Slide number 17 1 st order Linear Nonhomogeneous Initial](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-17.jpg)
Example n n n n Slide number 17 1 st order Linear Nonhomogeneous Initial value problem 2 nd order Linear Nonhomogeneous Boundary value problem
![Example n n n n Slide number 18 2 nd order Nonlinear Homogeneous Initial Example n n n n Slide number 18 2 nd order Nonlinear Homogeneous Initial](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-18.jpg)
Example n n n n Slide number 18 2 nd order Nonlinear Homogeneous Initial value problem 2 nd order Linear Homogeneous Initial value problem
![Solution Methods Direct Integration n n Slide number 19 This method works for Solution Methods - Direct Integration n n Slide number 19 This method works for](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-19.jpg)
Solution Methods - Direct Integration n n Slide number 19 This method works for equations where the RHS does not depend on the unknown: The general form is:
![Direct Integration n n Slide number 20 y is called the unknown or dependent Direct Integration n n Slide number 20 y is called the unknown or dependent](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-20.jpg)
Direct Integration n n Slide number 20 y is called the unknown or dependent variable; t is called the independent variable; “solving” means finding a formula for y as a function of t; Mostly we use t for time as the independent variable but in some cases we use x for distance.
![Direct Integration Example Find the velocity of a car that is accelerating from Direct Integration – Example Find the velocity of a car that is accelerating from](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-21.jpg)
Direct Integration – Example Find the velocity of a car that is accelerating from rest at 3 ms-2: If the car was initially at rest we have the condition: Slide number 21
![Bending of a beam Example A beam under uniform load Beam theory gives Bending of a beam - Example A beam under uniform load Beam theory gives](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-22.jpg)
Bending of a beam - Example A beam under uniform load Beam theory gives the governing equation: with boundary conditions: (pinned ends) Slide number 22
![Bending of a beam Solution Step 1 Integrate Step 2 Integrate Bending of a beam - Solution § Step 1: Integrate § Step 2: Integrate](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-23.jpg)
Bending of a beam - Solution § Step 1: Integrate § Step 2: Integrate again to obtain the general solution: Slide number 23
![Bending of a beam Solution Step 3 Use the boundary conditions to Bending of a beam - Solution § Step 3: Use the boundary conditions to](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-24.jpg)
Bending of a beam - Solution § Step 3: Use the boundary conditions to obtain the particular solution. § Step 4: Substitute back the values for A and B Slide number 24
![Solution Methods Separation The separation method applies only to 1 st order ODEs Solution Methods - Separation The separation method applies only to 1 st order ODEs.](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-25.jpg)
Solution Methods - Separation The separation method applies only to 1 st order ODEs. It can be used if the RHS can be factored into a function of t multiplied by a function of y: Slide number 25
![Separation General Idea First Separate Then integrate LHS with respect to y RHS Separation – General Idea First Separate: Then integrate LHS with respect to y, RHS](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-26.jpg)
Separation – General Idea First Separate: Then integrate LHS with respect to y, RHS with respect to t. Slide number 26
![Separation Example Separate Now integrate Slide number 27 Separation - Example Separate: Now integrate: Slide number 27](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-27.jpg)
Separation - Example Separate: Now integrate: Slide number 27
![Cooling of a cup of coffee Amount of heat in a cup of coffee Cooling of a cup of coffee Amount of heat in a cup of coffee:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-28.jpg)
Cooling of a cup of coffee Amount of heat in a cup of coffee: heat volume density specific heat temperature Heat balance equation (words): n Rate of change of heat = heat lost to surrounding air Slide number 28
![Cooling of a cup of coffee Newtons law of cooling n Heat lost to Cooling of a cup of coffee Newton’s law of cooling: n Heat lost to](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-29.jpg)
Cooling of a cup of coffee Newton’s law of cooling: n Heat lost to the surrounding air is proportional to temperature difference between the object and the air n The proportionality constant involves the surface area multiplied by a heat tranfer coefficient Heat balance equation (maths) : Slide number 29
![Cooling of a cup of coffee Substitute in rearrange where Now we solve the Cooling of a cup of coffee Substitute in rearrange where Now we solve the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-30.jpg)
Cooling of a cup of coffee Substitute in rearrange where Now we solve the equation together with the initial condition: Slide number 30
![Cooling of a cup of coffee Solution Step 1 Separate Step 2 Cooling of a cup of coffee Solution § Step 1: Separate § Step 2:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-31.jpg)
Cooling of a cup of coffee Solution § Step 1: Separate § Step 2: Integrate Make explicit in unknown T where Slide number 31
![Cooling of a cup of coffee Solution Step 3 Use Initial Condition Cooling of a cup of coffee Solution § Step 3: Use Initial Condition §](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-32.jpg)
Cooling of a cup of coffee Solution § Step 3: Use Initial Condition § Step 4: Substitute back to obtain final answer Slide number 32
![Solution Methods Integrating Factor The integrating factor method is used for nonhomogeneous linear Solution Methods - Integrating Factor The integrating factor method is used for nonhomogeneous linear](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-33.jpg)
Solution Methods - Integrating Factor The integrating factor method is used for nonhomogeneous linear 1 st order equations The basic ideas are: § Collect all the terms involving y and on the left hand side of the equation. § Combine them together as the derivative of a single function of y and t. § Solve by direct integration. The cunning trick is that step 2 cannot usually be done unless you first multiply the whole equation by an integrating factor. Slide number 33
![Integrating Factor Example There are several ways to solve this problem but we Integrating Factor – Example There are several ways to solve this problem but we](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-34.jpg)
Integrating Factor – Example There are several ways to solve this problem but we will use it to demonstrate the integrating factor method. To understand the integrating factor method you must be very familiar with the formula for the derivative of a product: Slide number 34
![Integrating Factor Example Product Rule The basic idea is that if we multiply Integrating Factor – Example Product Rule: The basic idea is that if we multiply](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-35.jpg)
Integrating Factor – Example Product Rule: The basic idea is that if we multiply the ODE by the correct function (an integrating factor) we can make the LHS of the ODE look like the RHS of the product rule. Slide number 35
![Integrating Factor Example We will look ahead use the answer and show it Integrating Factor – Example We will look ahead, use the answer and show it](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-36.jpg)
Integrating Factor – Example We will look ahead, use the answer and show it is derived later. For our ODE the integrating factor is Thus the ODE becomes: Now the LHS of this equation looks like the RHS of the Product Rule with: Slide number 36
![Integrating Factor Example We can rewrite the equation as or Now we can Integrating Factor – Example We can rewrite the equation as: or Now we can](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-37.jpg)
Integrating Factor – Example We can rewrite the equation as: or Now we can use direct integration Do not forget C, the constant of integration! Slide number 37
![Integrating Factor Example Rearrange to make this explicit in y Now use the Integrating Factor – Example Rearrange to make this explicit in y Now use the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-38.jpg)
Integrating Factor – Example Rearrange to make this explicit in y Now use the initial condition to calculate C Substitute back to obtain the final solution Slide number 38
![How do we calculate the integrating factor n n Let us now pretend we How do we calculate the integrating factor? n n Let us now pretend we](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-39.jpg)
How do we calculate the integrating factor? n n Let us now pretend we do not know what the integrating factor should be Call it Φ and use it to multiply the ODE from the previous example § To make the LHS of this equation look like the RHS of the Product Rule we must choose Slide number 39
![How do we calculate the integrating factor n Then the ODE becomes Now How do we calculate the integrating factor? n Then the ODE becomes § Now](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-40.jpg)
How do we calculate the integrating factor? n Then the ODE becomes § Now using the product rule in reverse the LHS can be written as a single term (a very clever trick) Slide number 40
![How do we calculate the integrating factor n Now we can integrate once we How do we calculate the integrating factor? n Now we can integrate once we](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-41.jpg)
How do we calculate the integrating factor? n Now we can integrate once we know Φ n We can separate to find Φ § The convention is to put A = 1. It appears in every term of the ODE, and therefore can be divided out. This gives the integrating factor: Slide number 41
![Finding the integrating factor in general n Given the general form of a nonhomogeneous Finding the integrating factor in general n Given the general form of a nonhomogeneous](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-42.jpg)
Finding the integrating factor in general n Given the general form of a nonhomogeneous 1 st order equation: n How do we use the integrating factor method to find a y? Slide number 42
![Finding the integrating factor in general n Step 1 Multiply by Φ n Step Finding the integrating factor in general n Step 1: Multiply by Φ: n Step](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-43.jpg)
Finding the integrating factor in general n Step 1: Multiply by Φ: n Step 2: Compare with the RHS of the Product Rule and set up equation for Φ: n Step 3: Use separation to solve for Φ: Slide number 43
![Finding the integrating factor in general n Step 4 Combine terms on the LHS Finding the integrating factor in general n Step 4: Combine terms on the LHS:](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-44.jpg)
Finding the integrating factor in general n Step 4: Combine terms on the LHS: n Step 5: Integrate: n Step 6: Divide by to make explicit in y : n Step 7: Use the initial conditions to evaluate C : Slide number 44
![Finding the integrating factor in general Notes n After you have been through the Finding the integrating factor in general Notes: n After you have been through the](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-45.jpg)
Finding the integrating factor in general Notes: n After you have been through the process a few times then skip some of the steps. For example you can remember the formula for the integrating factor, you do not have to rederive it every time. n In Slide number 45 principle this process can be used to solve any linear nonhomogeneous 1 st order ODE but some of the details may be tricky or impossible. Both the integrals and may be impossible to evaluate.
![Solving an example using the integrating factor method Step 1 Put the ODE into Solving an example using the integrating factor method Step 1: Put the ODE into](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-46.jpg)
Solving an example using the integrating factor method Step 1: Put the ODE into the general form: The ODE is already in that form! Step 2: Find the integrating factor: Slide number 46
![Solving an example using the integrating factor method Step 3 Multiply by the integrating Solving an example using the integrating factor method Step 3: Multiply by the integrating](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-47.jpg)
Solving an example using the integrating factor method Step 3: Multiply by the integrating factor: Step 4: Use the reverse Product Rule: Step 5: Integrate and make explicit in y: Slide number 47
![Solving an example using the integrating factor method Step 6 Use the initial conditions Solving an example using the integrating factor method Step 6: Use the initial conditions](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-48.jpg)
Solving an example using the integrating factor method Step 6: Use the initial conditions to find the exact solution: Step 7: Substitute back into the original equation: Slide number 48
![Exponential substitution n n The exponential trial or guessing method can be used for Exponential substitution n n The exponential trial or guessing method can be used for](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-49.jpg)
Exponential substitution n n The exponential trial or guessing method can be used for solving linear constant coefficient homogeneous differential equations. The basic trial for the solution of the ODE is: Slide number 49
![Characteristic Equations n gives the differentials n An algebraic characteristic equation comes from substituting Characteristic Equations n gives the differentials n An algebraic characteristic equation comes from substituting](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-50.jpg)
Characteristic Equations n gives the differentials n An algebraic characteristic equation comes from substituting in for y and its derivatives and cancelling out Slide number 50
![Exponential Trial Example Try n Cancelling out equation n Substituting this back into Exponential Trial - Example Try n Cancelling out equation n Substituting this back into](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-51.jpg)
Exponential Trial - Example Try n Cancelling out equation n Substituting this back into Slide number 51 gives the characteristic gives
![Solving Guide General Form Slide number 52 Description Solving Method 1 st Solving Guide General Form Slide number 52 Description Solving Method § § 1 st](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-52.jpg)
Solving Guide General Form Slide number 52 Description Solving Method § § 1 st or higher order RHS does not depend on the unknown y Direct Integration § 1 st order only Separation § § § 1 st order nonhomogeneous linear equation Integrating Factor Method
![Solving Guide General Form Slide number 53 Description Solving Method 2 nd Solving Guide General Form Slide number 53 Description Solving Method § § 2 nd](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-53.jpg)
Solving Guide General Form Slide number 53 Description Solving Method § § 2 nd order or higher Exponential trial. homogeneous See Module 2 linear equation constant coefficients § § 2 nd order or higher nonhomogeneous linear equation constant coefficients Problems like this can be solved for some types of f. Covered in Modules 3 and 5
![Solving Guide General Form Slide number 54 Description Solving Method 2 nd Solving Guide General Form Slide number 54 Description Solving Method § § 2 nd](https://slidetodoc.com/presentation_image_h/9c13bb658381be22a8e152970b2119cc/image-54.jpg)
Solving Guide General Form Slide number 54 Description Solving Method § § 2 nd order or higher Can only solve a nonhomogeneous few ‘special’ linear equation problems variable coefficients Not Covered in MM 2 § § 2 nd order Function f contains t, y and y’ terms all mixed up Generally can’t be solved analytically (see Module 4 for numerical methods)
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