First Oder Differential Equations

[ambanks04]

Homework Statement

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.

Homework 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

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.

 

[jackmell]

Homework Equations

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

.

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.

 

[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.