Question

Differential equation

Answer 1

By using the substitution \[z=\frac{ 1 }{ y^2}\]
Solve the differential equation
\[x^2\frac{ dy }{ dx} +xy = y^3\]

Answer 2

Ooook ok this looks familiar.
Something to do with Bernoulli, right?

So ummm... in order for this substitution to work for us,
let's try to get a 1/y^2 somewhere in our equation.
How bout dividing through by y^3?\[\large\rm x^2\frac{1}{y^3}y'+\frac{x}{y^2}=1\]

Answer 3

And let's take a derivative of our substitution with respect to x,\[\large\rm z=\frac{1}{y^2}\qquad\to\qquad z'=-\frac{2}{y^3}y'\]This looks very similar to our first block of junk in our differential equation, ya?

Answer 4

yep

Answer 5

\[\large\rm x^2\color{orangered}{\frac{1}{y^3}y'}+\frac{x}{\color{orangered}{y^2}}=1\]Using our z and z' we can replace all of this orange stuff somehow.
Just need to make a small adjustment to the first part.
Understand? :O

Answer 6

oh ok