Ordinary Differential Equations Basics w A differential equation

Ordinary Differential Equations (Basics) w A differential equation is an algebraic equation that contains some derivatives: • • Recall that a derivative indicates a change in a dependent variable with respect to an independent variable. In these two examples, y is the dependent variable and t and x are the independent variables, respectively.

Why study differential equations? • Many descriptions of natural phenomena are relationships (equations) involving the rates at which things happen (derivatives). • Equations containing derivatives are called differential equations. • Ergo, to investigate problems in many fields of science and technology, we need to know something about differential equations.

Why study differential equations? • Some examples of fields using differential equations in their analysis include: — Solid mechanics & motion — heat transfer & energy balances — vibrational dynamics & seismology — aerodynamics & fluid dynamics — electronics & circuit design — population dynamics & biological systems — climatology and environmental analysis — options trading & economics

Examples of Fields Using Differential Equations in Their Analysis

Differential Equation Basics • The order of the highest derivative in a differential equation indicates the order of the equation.

Simple Differential Equations A simple differential equation has the form Its general solution is

Simple Differential Equations Ex. Find the general solution to

Simple Differential Equations Ex. Find the general solution to

Exercise: (Waner, Problem #1, Section 7. 6) Find the general solution to

Example: Motion A drag racer accelerates from a stop so that its speed is 40 t feet per second t seconds after starting. How far will the car go in 8 seconds? Given: Find:

Solution: Apply the initial condition: s(0) = 0 The car travels 1280 feet in 8 seconds

Exercise: (Waner, Problem #11, Section 7. 6) Find the particular solution to Apply the initial condition: y(0) = 1

Separable Differential Equations A separable differential equation has the form Its general solution is Example: Separable Differential Equation Consider the differential equation a. Find the general solution. b. Find the particular solution that satisfies the initial condition y(0) = 2.

Solution: a. Step 1 — Separate the variables: Step 2 — Integrate both sides: Step 3 — Solve for the dependent variable: This is the general solution

Solution: (continued) b. Apply the initial (or boundary) condition, that is, substituting 0 for x and 2 for y into the general solution in this case, we get Thus, the particular solution we are looking for is

Exercise: (Waner, Problem #4, Section 7. 6) Find the general solution to