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1.

[x^2-2y^2]dx + xy dy = 0

xy dy = [2y^2 - x^2]dx

dy/dx = 2y/x - x/y

dy/dx - 2/xy = -x(y^-1)

Multiply by y

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

Let v= y^2

Then

dv/dx - 4 (x^-1) V = -2x

Since p(x) = -4/x, e ^integrate -(x) = x^-4

Multiply by x^-4

we have d (x^-4 v) /dy = -2x * x^-4 = -2(x^-3)

Integrate -2(x^-3)

we have x^-2 + C

Hence x^-4 * v = x^-2 + C

since v=y^2,

x^4*y^2 = x^-2 + C

y^2 = x^2 + C(x^4)

y^2-x^2 = C(x^4)

x^4 = C^-1 (y^2 - x^2)

But the general answer is x^4 = C(y^2 - x^2). What did I do wrong?

2. y dx + [x^2 - x] dy = 0

y dx = [x - x^2] dy

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

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

Multiply by x^-2

x^-2 (dx/dy) - y^-1 * x^-1 = - (y^-1)

Let v = x^-1 then dv/dx = -(x^-2)(dx/dy)

then dx/dy + y^-1*v = y^-1

Then p(y) = y^-1

Calculate e^integrate p(y) we have y

so multipl by y

y (dv/dy) = y* y^-1* v = y^-1 * y

d (y*v)/dy = 1

integrate 1 and we have

y*v = y + C

since v = x^-1

y*x^-1 = y + C

y = yx + Cx

y-yx = Cx

y(1-x) = Cx But the general answer is y(x+1) = Cx

Please trust me I tried everything I could think of to fix the problems, but I couldn't. Every time I redo the problems, I got the same answers. With my knowledge of differential equation (I have just started 2 weeks ago), I am out of ideas).

I also have posted a question regarding different problem which I could not solve. I would appreciate it if you take a look at that question and instruct me how I should solve them (some people tried to help me but I still cannot get it). Right now I don't know either how to obtain IF from f(xy)ydx + f(xy)x dy = 0 equations nor change the form to dy/dx + p(x)y = c

Thank you.