Using substiution of the form v=1/y^k to solve 1st order DE

[useruseruser]

$$xy'+3y+4y^3=0$$
use substitution of the form $$v=1/y^k$$ for some positive integer k. Choose a value of k that cancels the y^3 term.

 

[HallsofIvy]

$$xy'+3y+4y^3=0$$
use substitution of the form $$v=1/y^k$$ for some positive integer k. Choose a value of k that cancels the y^3 term.

Well, have you tried at all? There's not a whole lot of deep thinking involved! Just do exactly what they tell you to do. If [tex]v= 1/y^k[/itex], then $y= 1/v^{1/k}$. Find the derivative of y and plug it and y into the equation. There should be a single value of k that simplifies the equation.[/tex]