Question

Another change of variable ODE problem...

Answer 1

Really struggling with these questions I can't see what I am doing wrong....I know I've asked similar/identical questions like this before, but this is one is not working out for me at all.

Answer 2

get dz/dt by itself and then integrate

Answer 3

dx/dt i mean

Answer 4

once you have x you can figure out what z(t) is

Answer 5

I have to get the answer in the order it asks.

I hate this question its more of a puzzle more than anything.

Answer 6

hold on let me try my way

Answer 7

@lalaly any ideas also? You've helped me on similar questions!

Answer 8

@amistre64 Any ideas also? This one is driving me insane >.<

Answer 9

@TuringTest Any ideas?

Answer 10

x(t)=t z(t)
\[\frac{dx(t)}{dt}=z(t)+t \frac{dz(t)}{dt}\]plugging it in the equaion
\[t^2z(z+tz')=2t^3 e^t+te^t(zt)^2+z^2t^2\]\[t^2z^2+t^3zz'=2t^3e^t+t^3z^2e^t+z^2t^2\]\[t^3zz'=2t^3e^t+t^3z^2e^t+z^2t^2-t^2z^2\]\[t^3zz'=2t^3e^t+t^3z^2e^t\]divide by t^3
\[zz'=2e^t+z^2e^t\]so\[z \frac{dz}{dt}=e^t(2+z^2)\]