As you said, make the substiution
then
so
Integrate both sides and then substitute for z.
Hi
I don't understand how solve the homogenous equation
I set z=y/x therefore dy/dx=(dz/dx)x + z
however i wasn't able to fully solve it
i have attached two questions
thanks
For the second.
dy/dx = 1/x[y + (x)/sin(y/x)]
= y/x + [1/sin(y/x)]
let v = y/x, y = vx, dy/dx = (x)dv/dx + v
(x)dv/dx + v = v + 1/sin(v)
(x)dv/dx = 1/sin(v)
sin(v)dv = 1/x dx
-cos(v) = ln|x| + C1
cos(v) = -ln|x| + C1
v = cos^-1(C1-ln|x|)
y = xcos^-1(C1-ln|x|)
If you don't believe me check this out.
x*y*y' - y(y + x/sin(y/x)) = 0 - Wolfram|Alpha
Also, here is a link that shows you how to show that a differential equation is homogeneous Homogeneous Differential Equations: Example 2