MAT 1228 Series and Differential Equations Section 3.7 Nonlinear Equations

HW

Preview Look at 2 types of common Nonlinear Second Order DE. Technique: Reduction of Order + Chain Rule

Recall: Second Order Linear D.E.

Second Order Nonlinear D.E. Second Order DE not in the form of Examples:

Second Order Nonlinear D.E. In practice, physical systems are better modeled by nonlinear DE.

Example 1 (Pendulum) When the angle is small, the motion can be modeled by

Second Order Nonlinear D.E. In practice, physical systems are better modeled by nonlinear DE. In general, difficult to solve analytically (give explicit or implicit solutions). Some common cases may be solved by specific techniques.

Case I Dependent Variable is Missing. Assume General Form Example

Example 2

Depends on the sign of the integration constant, there are 3 possible form of solutions

Example 2 Suppose additional Initial conditions:

Case II Independent Variable is Missing. Assume General Form Example

Example 3 First attempt…

Example 3 Second attempt…

Last Word… You may encounter one of these DE in E&M. Remember how to solve it or remember where to look up in the reference.