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## Homework Statement

I have differential equation;] this is equation from central force thing in Lagrange mechanics, you know, you know, its second order hahaha:D

[tex]y^3y^{\prime\prime}=ay+b[/tex]

## Homework Equations

I will using method of making a first degree of this equation

## The Attempt at a Solution

I have

[tex]\frac{\mbox{d}y}{\mbox{d}x}=u(y)[/tex]

[tex]\frac{\mbox{d}^2y}{\mbox{d}x^2}=\frac{\mbox{d}u}{\mbox{d}x}=\frac{\mbox{d}u}{\mbox{d}y}\frac{\mbox{d}y}{\mbox{d}x}=u\frac{\mbox{d}u}{\mbox{d}y}[/tex]

and now I enter this new equation to before:

[tex]uy^3\frac{\mbox{d}u}{\mbox{d}y}=ay+b[/tex]

[tex]u\frac{\mbox{d}u}{\mbox{d}y}=\frac{ay+b}{y^3}[/tex]

[tex]u^2=-2\frac{a}{y}-4\frac{b}{y^2}+C[/tex]

[tex]u=\frac{\sqrt{Cy^2-2ay-4b}}{y}[/tex]

[tex]\frac{y}{\sqrt{Cy^2-2ay-4b}}\frac{\mbox{d}y}{\mbox{d}x}=1[/tex]

someone who is good in mathematic, please tell me if this calculations was good and what can I do now, I was thinking about do this with area functions, but not sure, please tell me simplest way to solve it;] thank you!