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

Question

2122 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-09-19T12:00:27

$y = xln|x| + Cx$

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:

$y = xln|x| + Cx$

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