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

,

but two cases are worthy of discussion:

**(1)**-
**Equations with the***y*missingLet

*v*=*y*'. Then the new equation satisfied by*v*isThis 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 haveThis is a first order linear differential equation. Its resolution gives

Since

*v*(1) = 1, we get . Consequently, we haveSince

*y*'=*v*, we obtain the following equation after integrationThe condition

*y*(1) = 2 gives . Therefore, we haveNote that this solution is defined for

*x*> 0.

**(2)**-
**Equations with the***x*missingLet

*v*=*y*'. Sincewe get

This is again a first order differential equation. Once

*v*is found then we can get*y*throughwhich 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 toThis a first order separable differential equation. Its resolution gives

Since , we get

*y*' = 0 orSince 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 areNote 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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