differential eqn substitution

differential eqn substitution

hey how do i solve
xy’ + y = exp(xy) using v=xy ?

ive done so far:
d/dx (xy) = exp(xy)
integrate wrt x:
xy = ∫ exp (xy) dx
then with v = xy:
v = ∫ exp(v) dx

but im not sure how you integrate exp(v) since v(x) = xy(x) so it will depend on what y(x) actually is yk. apparently it can be solved for y explicitly but i have no idea where to go from here. thanks in advance

Reply 1

mqb2766

Original post by yuhyuh726329

Use the substitution to get (as you have)
v' = e^v
Then you could divide through by e^v and you should recognise how you can integrate that as its just seperation of variables. Then finally replace v with xy and rearrange.