# First Oder Differential Equations

1. Oct 9, 2011

### ambanks04

1. The problem statement, all variables and given/known data
Solve the DE subject to y(1)=0. Forconvenience let k=v_r/v_s.

First let me apologize for the way it is written but I don't know how to use software like other on other posts I see.

2. Relevant equations
(dy/dx)= (v_s*y - v_r*sqrt(x^2+y^2)) / (v_s*x)

this may be more visual

dy v_s*y - v_r*sqrt(x^2+y2)
--=--------------------------
dx v_s*x

3. The attempt at a solution
I have about ten steps where I try to solve this problem. It is from the textbook :A First Course in Differential Equations" by Dennis Zill in section 3.2 #27. I think it is a seperable equation but I cannot get it to a form where I can solve the intial value problem. It is a horrible feeling to be stuck on the algebra of all things. Help is appreiciated. I need to learn how to do this so I can pass the midterm.

2. Oct 9, 2011

### jackmell

Keep in mind the term:

$$\sqrt{x^2+y^2}$$

is homogeneous of degree 1. The other terms are also homogeneous of degree 1. Also, we use the Latex programming language to format math. You can press the Quote button to see the various constructs enclosed in tex brackets.

3. Oct 9, 2011

### hunt_mat

So you have a differential equation of the form:
$$\frac{dy}{dx}=\frac{ay-b\sqrt{x^{2}+y^{2}}}{ay}$$
I would think about making the substitution:
$$y(x)=xv(x)$$
and see where that leads you.