Nonlinear Second Order Differential Equations

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

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but two cases are worthy of discussion:

(1)
Equations with the y missing

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

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

Example 1: Find the solution of

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

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

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Since v(1) = 1, we get tex2html_wrap_inline112 . Consequently, we have

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

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The condition y(1) = 2 gives tex2html_wrap_inline122 . Therefore, we have

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Note that this solution is defined for x > 0.

(2)
Equations with the x missing

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Let v = y'. Since

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we get

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

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

Example 2: Find the general solution of the equation

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

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This a first order separable differential equation. Its resolution gives

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Since tex2html_wrap_inline148 , we get y' = 0 or

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

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where C and tex2html_wrap_inline158 are two constants. All the solutions of our initial equation are

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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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Author: Mohamed Amine Khamsi

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