• Support PF! Buy your school textbooks, materials and every day products Here!

Separate Variables Differential Eq. of Cubic Power

  • Thread starter knowLittle
  • Start date
  • #1
307
0

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.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

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:
  • #3
307
0
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.
 
  • #4
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?
 
  • #6
307
0
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.
 
  • #7
33,636
5,296
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.
 

Related Threads on Separate Variables Differential Eq. of Cubic Power

Replies
8
Views
854
Replies
4
Views
3K
Replies
2
Views
2K
Replies
5
Views
1K
Replies
5
Views
1K
Replies
3
Views
2K
Replies
3
Views
3K
Replies
22
Views
4K
Replies
8
Views
3K
Replies
4
Views
957
Top