MAT 1228Series and Differential

Equations

Section 3.7

Nonlinear Equations

http://myhome.spu.edu/lauw

HW

Preview

Look at 2 types of common Nonlinear Second Order DE.

Technique:

Reduction of Order + Chain Rule

Recall: Second Order Linear D.E.

)()()()(

0)()()(

xGyxRyxQyxP

sHomogeneouNon

yxRyxQyxP

sHomogeneou

Second Order Nonlinear D.E.

( ) ( ) ( ) 0

( ) ( ) ( ) ( )

P x y Q x y R x y

P x y Q x y R x y G x

Second Order DE not in the form of

Examples:

22 0

2 0

y x y

y yy

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

l

02

2

l

g

dt

d

2

2sin 0

d g

dt l

3 5 7

sin3! 5! 7!

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

( , , ) 0F x y y ( )y x

22 0y x y

Example 2

22 0y x y

Example 2

22 0y x y

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

2

1y dx

x C

Example 2

22 0y x y

Suppose additional Initial conditions:

2

1y dx

x C

(0) 1, (0) 0y y

Case II

Independent Variable is Missing. Assume General Form Example

( , , ) 0F y y y ( )y x

2 0y yy

Example 3 First attempt…

2 0y yy (0) 1, (0) 1y y

Example 3 Second attempt…

2 0y yy (0) 1, (0) 1y y

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.