# Nonlinear Second Order Differential Equations

In general, little is known about nonlinear second order differential equations

,

but two cases are worthy of discussion:

(1)
Equations with the y missing

Let v = y'. Then the new equation satisfied by v is

This is a first order differential equation. Once v is found its integration gives the function y.

Example 1: Find the solution of

Solution: Since y is missing, set v=y'. Then, we have

This is a first order linear differential equation. Its resolution gives

Since v(1) = 1, we get . Consequently, we have

Since y'=v, we obtain the following equation after integration

The condition y(1) = 2 gives . Therefore, we have

Note that this solution is defined for x > 0.

(2)
Equations with the x missing

Let v = y'. Since

we get

This is again a first order differential equation. Once v is found then we can get y through

which is a separable equation. Beware of the constants solutions.

Example 2: Find the general solution of the equation

Solution: Since the variable x is missing, set v=y'. The formulas above lead to

This a first order separable differential equation. Its resolution gives

Since , we get y' = 0 or

Since this is a separable first order differential equation, we get, after resolution,

,

where C and are two constants. All the solutions of our initial equation are

Note that we should pay special attention to the constant solutions when solving any separable equation. This may be source of mistakes...

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