Cauchy-Euler

The second order cauchy-euler equation has the form

ax^2{{d^2y}/{dx^2}}+bx{{dy}/{dx}}+cy=h(x)

Where a, b and c are constants. The tansformation x=e^t (x>0) reduces the equation to a linear ordinary differential equation. Using the chain rule

x=e^t

{{dx}/{dt}}=e^t=x

{{dy}/{dt}}={{dy}/{dx}}{{dx}/{dt}}=x{{dy}/{dx}}

{{d^2y}/{dt^2}}={d/{dt}}({dy}/{dt})={d/{dt}}(x{dy}/{dx})

{{d^2y}/{dt^2}}={{dx}/{dt}}{{dy}/{dx}}+x{d/{dt}}({dy}/{dx})

{{d^2y}/{dt^2}}={{dy}/{dt}}+x{{dx}/{dt}}{d/{dx}}({dy}/{dx})

{{d^2y}/{dt^2}}={{dy}/{dt}}+x{{dx}/{dt}}{d^2y}/{dx^2}

{{d^2y}/{dt^2}}={{dy}/{dt}}+(x^2){d^2y}/{dx^2}

The transformation produces

x{{dy}/{dx}}={{dy}/{dt}}

(x^2){d^2y}/{dx^2}={{d^2y}/{dt^2}}-{{dy}/{dt}}

a{{d^2y}/{dt^2}}+(b-a){{dy}/{dt}}+cy=h(e^t)