If we try to use separation of variables to solve the IVP...?

Question

If we try to use separation of variables to solve the IVP

y' = (t^2 + 7) / (y^4) - 4y^3 , y(0) = 1 we would obtain a solution that could not be put in explicit form. But we can solve for

(y^5) / 5 - y^4 instead.....so do this....

((y^5) / 5) - y^4 = ????

Answer 1

not sure what you want?????
solution to this is
separate
[ y^4 - 4y^3 ] dy = [ t^2 +7 ] dt
y^5 / 5 - y^4 = t^3 / 3 + 7t +C
plug in t=0,y=1 to get C
1/5 - 1 = 0+0 + C
c = -4/5

y^5 / 5 - y^4 = t^3 / 3 + 7t - 4/5

you cannot solve for y=...

is this what you want?????

Answer 2

I dont think you can get to there from here.
You can do the following:
y'+ 4y^3 =( t^2+7)/y^4
y^4 y' + 4y^7 = (t^2+7)