x^2 * y' + 2 * x * y = e^x
Делим на х^2 левую и правую части.
y' + 2 * y/x = e^x/x^2
1) y' + 2 * y/x = 0
dy/dx + 2 * y/x = 0
dy/dx = -2 * y/x
dy/y = -2 * dx/x
int dy/y = -2 * int dx/x
ln |y| = -2 * ln |x| + C
y = C/x^2
2) y' + 2 * y/x = e^x/x^2 и y(x) = C(x)/x^2
Тогда y' = (C'(x) * x^2 - C(x) * 2x)/x^4 = C'(x)/x^2 - 2 * C(x)/x^3
Подставляем в уравнение
C'(x)/x^2 - 2 * C(x)/x^3 + 2 * C(x)/x^3 = e^x/x^2
C'(x)/x^2 = e^x/x^2 => C'(x) = e^x => C(x) = int e^x dx = e^x + C
Тогда y = (e^x + C)/x^2.