# What is the general solution of the differential equation # dy/dx = (x+y)/x #?

##### 1 Answer

Sep 19, 2017

# y = xln|x| + Cx #

#### Explanation:

We have:

# dy/dx = (x+y)/x # ..... [A]

If we use the suggested substitution,

# dy/dx = (v)(d/dx x) + (d/dx v)(x) #

# \ \ \ \ \ = (v)(1) + (dv)/dx x #

# \ \ \ \ \ = v + x(dv)/dx #

Substituting this result into the initial differential equation [A] we get:

# v + x(dv)/dx = (x+vx)/x #

# :. v + x(dv)/dx = 1+v #

# :. x(dv)/dx = 1 #

# :. (dv)/dx = 1/x #

Which has reduced the equation to a trivial First Order separable equation, which we can *"separate the variables"* to get:

# int \ dv = int \ 1/x \ dx #

And if we integrate we get:

# v = ln|x| + C #

Restoring the earlier substitution, we get:

# y/x = ln|x| + C #

Leading to the General Solution:

# y = xln|x| + Cx #