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.