Cauchy-Euler Diff. Eqn: Transformation & Solution

[Arman777]

Cauchy-Euler is a type of diff equation which is described by

$$a_0x^2(\frac {d^2y} {dx^2})+a_1x(\frac {dy} {dx})+a_2y=F(x)$$

The transformation of $x=e^t$ can solve the equation.

Now, in here I didnt understand how to transform $\frac {dy} {dx}$ to $\frac {dy} {dt}$.

it goes like this $\frac {dy} {dx}=\frac {dy} {dt} \frac {1} {x}$ and then I am stuck I should take another derivative but I couldn't do it somehow.

 

[Orodruin]

Let $dy/dt = f$ and apply the product rule for derivatives when differentiating $f/x$ wrt x.

 

[Arman777]

Let $dy/dt = f$ and apply the product rule for derivatives when differentiating $f/x$ wrt x.

Okay hmm

$\frac {dy} {dx}=\frac {f} {x}$
$\frac {d^2y} {dx^2}=\frac {df} {dx} \frac {1} {x}-\frac {f} {x^2}$

now I understand until here, but I didnt understand this term $\frac {df} {dx}=\frac {d^2y} {dt^2}\frac {1} {x}$ which its done in the book.

 

[Orodruin]

What is $df/dx$ in terms of $df/dt$?

 

[Arman777]

What is $df/dx$ in terms of $df/dt$?

$df/dx=(df/dt)(dt/dx)$

I understand it now, thanks