Chapter 2 Firstorder Differential Equations Chapter 1 Introduction

Chapter 2 First-order Differential Equations

Chapter 1: Introduction to Differential Equations Overview I. Separable variables II. Linear equations III. Exact Equations IV. Solution by substitutions

I. Separable variables Learning Objective At the end of this section you should be able to identify and solve a separable DE.

I. Separable variables Definition A first-order DE of the form is said to be separable or to have separable variables.

I. Separable variables Examples: is separable Indeed, is not separable Indeed, it cannot be put on the product form

I. Separable variables Remark: By dividing by , the separable DE, can be written in the form: where

I. Separable variables Method of solution: From the form : we have to integrated both sides. to obtain One-parameter family of implicit or explicit solutions.

I. Separable variables Examples:

I. Separable variables Examples:

I. Separable variables Examples: . So, the solution of the IVP:

I. Separable variables Examples: .

I. Separable variables Examples: Solution of IVP: .

I. Separable variables Exercise I: Solve the following DE by separation of variables: .

II. Linear equations Learning Objective At the end of this section you should be able to identify and solve a linear DE.

II. Linear Equations Definition A first-order DE of the form is said to be linear equation in the dependent variable .

II. Linear Equations Remark By dividing both sides by written in the standard form: , a linear equation can be

II. Linear Equations Definition Values of that will make points of the equation. Example: is a singular point. , are called singular

II. Linear Equations Definition The function factor. Remark: is defined as the integrating

II. Linear Equations Examples: Find the integrating factor for : 1)

II. Linear Equations Examples: Find the integrating factor for : 2) Singular point :

II. Linear Equations Method of solution 1) Write the standard form : 2) Find the integrating factor :

II. Linear Equations Method of solution 3) Multiply the Standard form by the integrating factor : Standard form :

II. Linear Equations Method of solution 4) Integrate both sides of the last equation :

II. Linear Equations Examples: Solve the following linear DE : 1)

II. Linear Equations Examples: 1)

II. Linear Equations Examples: 2)

II. Linear Equations Examples: 2) Valid on

II. Linear Equations Exercise-II: Solve the following linear DE :

III. Exact equations Learning Objective At the end of this section you should be able to identify and solve an exact ODE.

III. Exact equations Definition Consider a function of two variables: is the partial derivative of regarding ( is considered as a constant).

III. Exact equations Definition Consider a function of two variables: its differential is :

III. Exact equations Example:

III. Exact equations Examples:

III. Exact equations Definition A differential expression : is an exact differential if it corresponds to the differential of some function that means

III. Exact equations Example is an exact differential Indeed

III. Exact equations Definition A DE of the form: is an exact equation if the left side is an exact differential. In that case, the DE is equivalent to An implicit solution will be

III. Exact equations Example is an exact equation Indeed

III. Exact equations Theorem A necessary and sufficient condition that be an exact differential is .

III. Exact equations Example Indeed exact equation

III. Exact equations Method of solution Example: Step 1: Check exactitude ? . Exact DE there exists such that

III. Exact equations Method of solution Step 2: integrate regarding . constant for but not for

III. Exact equations Method of solution Step 3: Differentiate regarding .

III. Exact equations Method of solution Step 4: Integrate regarding .

III. Exact equations Method of solution Step 5: Solution An implicit solution is . An explicit solution is defined when

III. Exact equations Example: Solve the following ODEs: exact equation

III. Exact equations Example:

III. Exact equations Example: Implicit solution

III. Exact equations Example: exact equation

III. Exact equations Example:

III. Exact equations Example: Family of implicit solutions Solution of the IVP:

III. Exact equations Exercise-IIIa: Determine whether the given DE is exact. If it is, solve it. 1. 2. 3. 4. 5.

III. Exact equations Remark: make exact some non-exact DEs is non exact if where There are cases where the equation can be made exact. How?

III. Exact equations Remark: make exact some non-exact DE compute If the result is a function of the sole variable then find the integrating factor : :

III. Exact equations Remark: make exact some non-exact DE Multiply the DE by the integrating factor : Now Exact!

III. Exact equations Remark: make exact some non-exact DE Example

III. Exact equations Remark: make exact some non-exact DE Example Exact!

III. Exact equations Remark: make exact some non-exact DE If is NOT a function of the unique variable compute If the result is a function of the sole variable then find the integrating factor : :

III. Exact equations Remark: make exact some non-exact DE Example

III. Exact equations Remark: make exact some non-exact DE Example Exact!

III. Exact equations Exercise-IIIb: Solve the given D. E by finding an appropriate integrating factor. 1. 2. 3. 4.

IV. Solution by substitutions Learning Objective At the end of this section you should be able to solve Homogeneous and Bernoulli’s DEs.

IV. Solution by substitutions Bernoulli Equation Definition : A DE in the form real number is said a Bernoulli equation. where is a

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Equation Substitution:

IV. Solution by substitutions Bernoulli Equation

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Equation Example :

IV. Solution by substitutions Bernoulli Exercise IVb : Solve the given Bernoulli equation by using an appropriate substitution.

IV. Solution by substitutions Homogeneous DE Definition : If a function has the property that for some real number , then homogeneous function of degree. , is said to be a

IV. Solution by substitutions Homogeneous DE Example 1 : is a homogeneous function of degree 2.

IV. Solution by substitutions Homogeneous DE Example 2 : is a homogeneous function of degree 2/3.

IV. Solution by substitutions Homogeneous DE Example 3 : We can’t factorize by a power of is not homogeneous

IV. Solution by substitutions Homogeneous DE Example 4 : is homogeneous of degree 0

IV. Solution by substitutions Homogeneous DE Definition : A DE of the form homogeneous if both and functions of the same degree. is said to be are homogeneous

IV. Solution by substitutions Homogeneous DE Example : is homogeneous. Indeed and are homogeneous of degree 2

IV. Solution by substitutions Homogeneous DE Method of solution: 1) Homogeneity : and are homogeneous of degree 1 2) Substitution

IV. Solution by substitutions Homogeneous DE Method of solution:

IV. Solution by substitutions Homogeneous DE Method of solution:

IV. Solution by substitutions Homogeneous DE Method of solution: 2) Substitution (2 nd option)

IV. Solution by substitutions Homogeneous DE Method of solution: 2) Substitution

IV. Solution by substitutions Homogeneous DE Method of solution: 2) Substitution

IV. Solution by substitutions Homogeneous DE Exercise IVa : Solve the given homogeneous equation by using an appropriate substitution.

End Chapter 2