# Separate Variables Differential Eq. of Cubic Power

## Homework Statement

When possible express the general solution in explicit form.
Solve dy/dx =x^2 /(1+y^2)

## Homework Equations

This is a first order non-linear ordinary differential equation.

## The Attempt at a Solution

dy(1+y^2) = x^2 dx
Integration both sides returns:
y+ (y^3 )/3= (x^3)/3 +C
Now, I am aware that there is more than one solution for y involving imaginary numbers. Can someone help me in the next step or direct me to a site?
I have seen cubic solutions tutorial, but they involve equations of the form: Ax^3+Bx^2+Cx+D=0

Thank you.

Dick
Homework Helper

## Homework Statement

When possible express the general solution in explicit form.
Solve dy/dx =x^2 /(1+y^2)

## Homework Equations

This is a first order non-linear ordinary differential equation.

## The Attempt at a Solution

dy(1+y^2) = x^2 dx
Integration both sides returns:
y+ (y^3 )/3= (x^3)/3 +C
Now, I am aware that there is more than one solution for y involving imaginary numbers. Can someone help me in the next step or direct me to a site?
I have seen cubic solutions tutorial, but they involve equations of the form: Ax^3+Bx^2+Cx+D=0

Thank you.

I really don't think you want to solve for y. With an expression like that I'd just leave in the implicit form you already have, or maybe express x as a function of y instead. Don't try to use the cubic formula. It's a mess.

Last edited:
Yes, I am aware that it is kinda difficult to solve for 'y' and that's why I wanted to try it out. It involves imaginary numbers and many roots.
If someone can point me in the right direction, that would great.

Luckily, this is a seperable equation, which means you can rewrite it as
$$(1+y^2)dy = x^2dx.$$
Now, what can you do to get rid of those pesky differentials?

Thank you, tiny-tim. I usually use latex for big equations, but I thought it wouldn't be a big deal.

I was thinking that I could solve it like your wikipedia link... this will be interesting. Thanks.

Mark44
Mentor
Luckily, this is a seperable equation, which means you can rewrite it as
$$(1+y^2)dy = x^2dx.$$
Now, what can you do to get rid of those pesky differentials?
You really need to read through the thread. The OP has already done this and has gotten a solution.