Problem 02
$(x - 2y) \, dx + (2x + y) \, dy = 0$
Solution 02
$(x - 2y) \, dx + (2x + y) \, dy = 0$
Let
$y = vx$
$dy = v \, dx + x \, dv$
Substitute,
$(x - 2vx) \, dx + (2x + vx)(v \, dx + x \, dv) = 0$
$x \, dx - 2vx \, dx + 2vx \, dx + 2x^2 \, dv + v^2x \, dx + vx^2 \, dv = 0$
$x \, dx + 2x^2 \, dv + v^2x \, dx + vx^2 \, dv = 0$
$(x \, dx + v^2x \, dx) + (2x^2 \, dv + vx^2 \, dv) = 0$
$x(1 + v^2) \, dx + x^2(2 + v) \, dv = 0$