I have to solve this DE

((x+2)^2)y'' -2y=4

the hint says to get rid of the x+2 but I have no idea where to start from... do I have to replace x+2 with another variable or what? (Doh)

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- Feb 10th 2011, 01:45 AMlsoSecond order non homogeneous differential equation
I have to solve this DE

((x+2)^2)y'' -2y=4

the hint says to get rid of the x+2 but I have no idea where to start from... do I have to replace x+2 with another variable or what? (Doh) - Feb 10th 2011, 02:21 AMAckbeet
I would replace the $\displaystyle x+2$ with $\displaystyle t=x+2,$ and you get

$\displaystyle \dfrac{dy}{dt}=\dfrac{dy}{dx}\dfrac{dx}{dt}=\dfrac {dy}{dx}.$ Hence, the DE becomes

$\displaystyle t^{2}\,\dfrac{d^{2}y}{dt^{2}}-2y=4.$

This equation is a Cauchy-Euler equation.