Solving a Differential Equation: y'/(1+y'^2) = 2y^2 + C

[psid]

Homework Statement

How to solve the following DE:
$$\frac{1}{\sqrt{1+(dy/dx)^{2}}}=\frac{2y^{2}}{2}+C$$?

 

[CompuChip]

I suppose solving it for dy/dx might enable you to do a separation of variables...

I.e. (since you are posting this in advanced physics): write
dy/dx = f(y)
for some function f only depending on y; then integrate
dx = dy / f(y)
and invert to find y(x).

Granted, it's probably easier said than done, but you can give it a try.

 

[psid]

It is indeed separable. I get it into the following form, but don't know how to integrate
$$dx=\sqrt{\frac{((2/\gamma)y^{2}+C)^{2}}{1-((2/\gamma)y^{2}+C)^{2}}}dy$$

 

[FedEx]

It is indeed separable. I get it into the following form, but don't know how to integrate
$$dx=\sqrt{\frac{((2/\gamma)y^{2}+C)^{2}}{1-((2/\gamma)y^{2}+C)^{2}}}dy$$

This is an elegant problem.

Superb.

First: Let's try to make the equation a bit less horrendous.

Take $$\sqrt{1-((2/\gamma)y^{2}+C)^{2}} = t$$

Proceed with that. Simplify it well and then take

$$t= sin\theta$$

Simplify it and then use De moivre's theorem.

 

[FedEx]

May i know the name of the book.

 

[psid]

But the problem with this substitution is that there is a second power of y in the square root. Thus there will be a term including y for the expression for dt...