Homework Help: Challenging ODE

1. Jun 28, 2010

lewis198

I've been going through problems in a classic maths for physics book, and am stuck on a question.

I can't find the solution for this equation:

y'=[a/(x+y)]^2

I tried directly integrating it, using iterative techniques, and I know it is not homogeneous so I can't use substitution.

2. Jun 28, 2010

Dick

It doesn't have to be homogeneous for substitution to be useful. Try u=x+y.

3. Jun 29, 2010

lewis198

thanks for the reply. What is the criteria for a substitution?

4. Jun 29, 2010

Dick

The only criterion for a substitution is that it simplifies the equation in such a way that you can solve it. The equation in terms of u is separable.

5. Jun 29, 2010

HallsofIvy

Whatever works!

6. Jun 30, 2010

lewis198

Thanks again. I used algebraic long division and trigonometric substitution and got the following solution:

y-arctan((x+y)/a)=C

This is not a so called 'closed' solution. Is that ok? Is this solution even meaningful?

7. Jun 30, 2010

HallsofIvy

Yes, that's a perfectly good solution. In general, since dx/dy= 1/(dy/dx), for first order differential equations, the distinction between "independent variable" and "dependent variable" doesn't exist and it is seldom possible to solve for one variable "in terms of" the other.

I remember in my first "differential equations" course, early in the course, I arrived at a solution that involved an integral I just could not do. After a long time of trying everything I could think of, I finally looked in the back of the book. I found that the answer was given in terms of that integral! When you working with differential equations, don't expect everything to be simple!

8. Jun 30, 2010

Dick

You've got the right technique. I think you might be missing a factor of 'a' though.