Understanding Substitution in Differential Equations | Homework Help

[wimma]

Homework Statement

I'm reading a book where they do the following steps which I don't understand:
We have a DE:
b^2 * y'' = axy
put t = b^(-2/3) a ^(1/3) x
then somehow get (d^2 y)/dt^2 = ty
how?

Homework Equations

None.

The Attempt at a Solution

I tried messing with chain rule but got nowhere.

 

[HallsofIvy]

Homework Statement

I'm reading a book where they do the following steps which I don't understand:
We have a DE:
b^2 * y'' = axy
put t = b^(-2/3) a ^(1/3) x
then somehow get (d^2 y)/dt^2 = ty
how?

Yes, the "chain rule" is the way to go.

If t= b^{-2/3}a^{1/3}x then dt/dx= b^{-2/3}a^{1/3} and x= b^{2/3}a^{-1/3}t

\frac{dy}{dx}= \frac{dy}{dt}\frac{dt}{dx}= b^{-2/3}a^{1/3}\frac{dy}{dt}

Doing that again,
\frac{d^2y}{dx^2}= b^{-4/3}a^{2/3}\frac{d^2y}{dt^2}

Now, we have
b^{2- 4/3}a^{2/3}\frac{d^2y}{dt^2}= a^{1- 1/3}b^{-2/3}ty
b^{-2/3}a^{2/3}\frac{d^2y}{dt^2}= a^{2/3}b^{-2/3}ty

\frac{d^2y}{dt^2}= ty

Homework Equations

None.

The Attempt at a Solution

I tried messing with chain rule but got nowhere.