Differential Equations and Substitutions (Calc 2)

[lelandsthename]

Homework Statement

Solve xy' = y + xe^(y/x) using the substitution v=(y/x)

Homework Equations

Solving differential equations, substitution

The Attempt at a Solution

x (dy/dx) = y + xe^(y/x)

(dy/dx) = (y/x) + e^(y/x)

Substituting v=(y/x)

(dy/dx) = v + e^(v)

I do not know how to proceed from here. (There are so many variables that aren't x and y! Ahh) Any guidance would be greatly appreciated!

 

[Hootenanny]

You need to make the change of variable with the differential too. In other words you need to write

$$\frac{dy}{dx}$$

In terms of x and v. Note that v=v(x), that is, v is a function of x.

 

[HallsofIvy]

If v=y/x, then y= ??

and from that dy/dx= ??

 

[lelandsthename]

Hmm, ok, so dv = dy/dx? Somehow I still think I'm missing something. Shouldn't there be a dv/dx somewhere or something? I am just not seeing it =/

 

[Hootenanny]

Hmm, ok, so dv = dy/dx? Somehow I still think I'm missing something. Shouldn't there be a dv/dx somewhere or something? I am just not seeing it =/

You're on the right lines, but not quite there. As Halls says,

$$y = v(x)\cdot x$$

Now you need to find the first derivative of the above function with respect to x,

$$\frac{dy}{dx} =\ldots$$