Verified answer

Hola

1) x y´= y + x

x dy = y dx + x dx

x dy - y dx = x dx

(x dy - y dx)/x^2 = x dx / x^2 = dx/x

d (y/x) = d(ln(x))

y/x = k + ln(x)

y = k x + x ln(x)

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2) (x-y)dx+(x-y+1)dy=0

u = x - y

du = dx - dy

dy = dx - du

u dx + (u - 1) (dx - du) = 0

(2 u - 1) dx = (u - 1) du

2 dx = (u - (1/2) - (1/2)) du/(u - (1/2))

2 dx = du - (1/2) d( ln(u - (1/2)) )

2 x = u - (1/2) ln(u - (1/2) ) + C

2 x = 2 x - 2 y - (1/2) ln(x - y - (1/2)) + C

4 y + ln( x - y - (1/2)) = C'

3) y^2+x^2 / 2xy

Falta el diferencial, la igualdad....

4) dy/dx= x/y + y/x

u = y/x

y = u x

dy = u dx + x du

u dx + x du = ( (1/u) + u) dx

x du = dx/u

u du = dx/x

(1/2) u^2 = ln(x) + c

(1/2) (y/x)^2 = ln(x) + c

5) xy´-y=x^2e^x

(x y' - y)/x^2 = e^x

(y/x)' = e^x

y/x = e^x + c

y = x e^x + c x