# 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... [Differential Equations] [First Order D.E.] [Second Order D.E.]
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