For some reason, this is giving me fits. Im supposed to put it in the form: then use the substitution , so and
Here is the D.E.
Is it safe to make it like this: then then use the substitution?
if you multiply by dx you get:
xdy = ydx + sqrt(x^2 +y^2)dx
(y + sqrt(x^2+y^2))dx -xdy = 0
using dy = vdx + xdv => (vx + sqrt(x^2+v^2x^2)dx - x(vdx+xdv) = 0
(vx + x sqrt(1+v^2) -xv)dx - x^2dv = 0
(xsqrt(1+v^2))dx -x^2 dv = 0
which now separates 1/x dx = dv/sqrt(1+v^2)
Typically this is not how you would approach this problem--no need to put
in the form Mdx + Ndy = 0.
dy/dx = y/x + sqrt[1+(y/x)^2]
simply y = vx dy/dx = v + xdv/dx
v+ xdv/dx = v +sqrt(1+v^2)
dv/sqrt(1+v^2) = dx/x
By the way if dy/dx = f(y/x) we call it a differnetial equation with homogeneous coefficients which can be solved by letting y = vx