Solve dy/dx=(y+(x^2+y^2)^0.5)/x
dy/dx = (y/x) + √(x^2 + y^2)/x
dy/dx = (y/x) + √(1 + (y/x)^2) --------- (1)
Let v = y/x, or y = vx, then
dy/dx = v + x.dv/dx -------------------- (2)
Substituting for v = y/x and dy/dx, from (2), into (1)
v + x.dv/dx = v + √(1 + v^2)
x.dv/dx = √(1 + v^2)
Rearranging,
dv/√(1 + v^2) = dx/x
integrating,
int dv/√(1 + v^2) = int dx/x
arcsinh(v) = ln(x) + ln(k) = ln(kx)
v = sinh(ln(kx))
y/x = sinh(ln(kx))
Answer: y(x) = x.sinh(ln(kx))