Затем делаем замену t = y/x; y = tx; y' = dy/dx = t'*x + t
1) 2xyy' = x^2 + y^2
Делим на 2xy
y' = x/(2y) + y/(2x)
y' = 1/2*(x/y + y/x)
t'*x + t = 1/2*(t + 1/t) = t/2 + 1/(2t)
dt/dx*x = t/2 + 1/(2t) - t = 1/(2t) - t/2 = (1 - t^2)/(2t)
2t/(1 - t^2) dt = dx/x
Уравнение с разделенными переменными. Интегрируем:
ln |1 - t^2| = ln |x| + ln C = ln |Cx|
1 - t^2 = 1 - y^2/x^2 = Cx
y^2 = x^2 - Cx^3
[b]y = sqrt(x^2 - Cx^3)[/b]
2) (x + y)dx + xdy = 0
xdy = -(x + y)dx
dy/dx = -(x + y)/x = -(1 + y/x)
t'*x + t = -(1 + t) = -1 - t
dt/dx*x = -1 - 2t
dt/(1 + 2t) = -dx/x
1/2*ln |1 + 2t| = -ln |x| + ln С
ln sqrt(1 + 2y/x) = ln (C/x)
1 + 2y/x = (C/x)^2
[b]y = x/2*((C/x)^2 - 1)[/b]
3) x(x + 2y)dx + (x^2 - y^2)dy = 0
(x^2 - y^2)dy = -(x^2 + 2xy)dx
dy/dx = -(x^2 + 2xy)/(x^2 - y^2)
Делим числитель и знаменатель дроби на x^2:
dy/dx = -(1 + 2y/x)/(1 - y^2/x^2)
t'*x + t = -(1 + 2t)/(1 - t^2)
dt/dx*x = -(1 + 2t)/(1 - t^2) - t
dt/dx*x = -(1 + 2t + t - t^3)/(1 - t^2) = (t^3 - 3t - 1)/(1 - t^2)
(1 - t^2)/(t^3 - 3t - 1) dt = dx/x
Замена t^3 - 3t - 1 = z; dz = (3t^2 - 3) dt = -3(1 - t^2) dt
-1/3*dz / z = dx/x
1/3*ln |z| = - ln |x| + ln C
∛z = ∛(t^3 - 3t - 1) = C/x
t^3 - 3t - 1 = (C/x)^3
[b](y/x)^3 - 3(y/x) = (C/x)^3 + 1[/b]
Как из этого получить y, я не знаю, но думаю, этого достаточно.
4) y' = (x - y) / (x - 2y)
Делим числитель и знаменатель дроби на x:
y' = (1 - y/x) / (1 - 2y/x)
t'*x + t = (1 - t)/(1 - 2t)
dt/dx*x = (1 - t)/(1 - 2t) - t
dt/dx*x = (1 - t - t + 2t^2)/(1 - 2t) = (2t^2 - 2t + 1)/(1 - 2t)
(1 - 2t)/(2t^2 - 2t + 1) dt = dx/x
Замена 2t^2 - 2t + 1 = z; dz = 4t - 2 dt = -2(1 - 2t) dt
-1/2*dz / z = dx/x
1/2*ln z = -ln x + ln C
sqrt(z) = sqrt(2t^2 - 2t + 1) = C/x
2t^2 - 2t + 1 = (C/x)^2
[b]2(y/x)^2 - 2y/x = (C/x)^2 - 1[/b]