2. Lecture (25): Ordinary Differential Equations (1 of 2)  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.

3. 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.

4. 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

5. Examples of Fields Using Differential Equations in Their Analysis

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

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

8. Ex. Find the general solution to Simple Differential Equations

9. Ex. Find the general solution to Simple Differential Equations

10. Find the general solution to Exercise: (Waner, Problem #1, Section 7.6)

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

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

13. Find the particular solution to Exercise: (Waner, Problem #11, Section 7.6) Apply the initial condition: y(0) = 1

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

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

16. Solution: (continued) 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 b.b.

17. Find the general solution to Exercise: (Waner, Problem #4, Section 7.6)