Question

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

Answer 1

`y = xln|x| + Cx`

Explanation:

We have:

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

If we use the suggested substitution, `y=vx` then differentiating wrt `x` and applying the product rule we have:

`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`