Homogenous Differential Equations (Converting Back into Explicit Form

[checkmatechamp]

Homework Statement

x*e^(y/x) + y dx = xdy, y(1) = 0

Homework Equations

The Attempt at a Solution

To solve, I divide everything by x dx to put everything in terms of v.

e^v + v = dy/dx

dy/dx = x dv/dx + v

e^v + v = x dv/dx + v

e^v = x dv/dx

e^v / dv = x/dx

Flip both sides.

e^-v dv = 1/x dx

Integrate both sides

-e^-v = ln|x| + c

-e^(-y/x) = e^(ln|x| + c)

-e^(-y/x) = x * e^c

-e^(-y/x) = cx

ln(-e^(-y/x)) = ln(cx)

ln(e^(x/-y)) = ln(cx)

-x/y = ln(cx)

1/y = -ln(cx)/x

y = -x/(ln(cx))

Is that the correct explicit form? Would it make it easier if I used the initial condition to find c, and then attempted to put it in explicit form, or not?

 

[xaos]

-e^-v = ln|x| + c

-e^(-y/x) = e^(ln|x| + c)

this step does not follow. otherwise I think you're on the right track.