Separate Variables Differential Eq. of Cubic Power

[knowLittle]

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 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.

 

[knowLittle]

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.

 

[YawningDog27]

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?

 

[tiny-tim]

hi knowLittle! :

(try using the X² button just above the Reply box )

start by writing it y³ + 3y + (C - x³) = 0

then use http://en.wikipedia.org/wiki/Cubic_equation#Trigonometric_.28and_hyperbolic.29_method (with p = 1, q = C - x³)

 

[knowLittle]

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]

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.