Solve the given initial-value problem. The DE is homogeneous.

Question

Solve the given initial-value problem. The DE is homogeneous.

(x + ye^y/x) dx − xe^y/x dy = 0,    y(1) = 0

Answer 1

y = vx; dy/dx = xdv/dx + v; dy/dx = (x + ye^y/x)/(xe^y/x); v + xdv/dx = (x + vxe^v)/(xe^v); xdv/dx = (1 + ve^v)/e^v - v;

xdv/dx = e^-v; e^vdv = dx/x; e^v = lnixI + C; e^y/x = lnIxI + C; e^0/1 = ln1 + C; C = 1; e^y/x = lnIxI + 1;

y = xln(lnIxI + 1)