Homogeneous differential equations
Differential equations (introductory) · Mathematics
Study notes
Q: Solve dy/dx = (x^2 + y^2)/(xy). Homogeneous degree 0 (each term degree 2/2). Let y = vx: dy/dx = v + x dv/dx. v + x dv/dx = (x^2 + v^2 x^2)/(v x^2) = (1 + v^2)/v. x dv/dx = (1+v^2)/v - v = 1/v. Separate: v dv = dx/x. Integrate: v^2/2 = ln|x| + C. Back: y^2/(2x^2) = ln|x| + C, so y^2 = 2x^2(ln|x| + C).