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

\[x(t)=z(t)\cdot t \rightarrow x'(t)=\frac{dz}{dt}\cdot t+z\]

Answer 3

And after simplifications, you can \[z\cdot z'=e^{t}(2+z^2)\]

Answer 4

Ugh, stupid product rule mistake cheers @athe

Answer 5

everything happens

Answer 6

@athe Can you where I went wrong, still not working out

Answer 7

Can you see where*

Answer 8

Try to accurately substitute the expression:
\[x(t)=z(t)\cdot t, x'(t) = t\cdot \frac{dz}{dt}+z; \]

Answer 9

\[t \cdot x\frac{dx}{dt}=2\cdot t^3\cdot e^t +t\cdot e^t \cdot x^2 +x^2\]

Answer 10

\[t\cdot (z\cdot t)\cdot (t\cdot z' +z)=2t^3\cdot e^t +t\cdot e^t \cdot (z\cdot t)^2+(z\cdot t)^2\]

Answer 11

After simplification, you will \[z\cdot z'=e^t(2+z^2)\]