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.

(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




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