Classifying a Diff Eq: Linear vs. Non-Linear Techniques

[formulajoe]

x(dy/dx) = y*e^(x/y) - x

its either separable, linear, homogeneous, bernoulli or exact. only thing i can figure is that its linear.

how do i break it down to figure it out? the e^x/y is what's throwing me off.

 

[arildno]

It's not linear!
Divide by x.
Note that your right-hand side can now be written as some function g(y/x).

 

[formulajoe]

if i break it apart i get dy/dx = (y*e^(x/y) / x) - 1. i can see how it could be separable, but the e^x/y would still be there when i integrate. and that integral would be fairly impossible. i don't see what else it can be.

 

[arildno]

Introduce the variable:
$$v(x)=\frac{y(x)}{x}$$
We have then:
$$v'(x)=\frac{y'(x)}{x}-\frac{v}{x}$$
Or:
$$y'(x)=xv'(x)+v(x)$$
Hence, inserting this into your diff. eq., you have:
$$xv'(x)=ve^{\frac{1}{v}}-(1+v)$$
This is a separable equation (I wouldn't try solving it, though..)