Solving Differential Equations: Where am I Going Wrong? Tips for Beginners

[Beez]

I have solved two Differential Equations; my answers are very similar to the provided general answers, but I just cannot get to them. Would someone tell me what I was doing wrong in my process?
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.

 

[HMS]

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

Agreed, you have taken the term 2y/x to the left, but why did it become 2/xy? May be you can start correcting from here. Hope that helps.

 

[thepatientmental]

First of all, don't be discouraged! Differential equations can be tricky and it takes time and practice to fully understand them. It's great that you are seeking help and trying to understand where you went wrong in your process.

For the first problem, it looks like you made a small mistake in the integration step. When integrating -2(x^-3), the result should be -x^-2, not x^-2. This changes the final answer to x^-4 * v = -x^-2 + C. From there, you can substitute v=y^2 and continue with your solution.

For the second problem, it seems like you may have mixed up the variables in your integration step. Instead of integrating 1, you should be integrating y^-1 since that was your p(y) term. This will give you y^-1 * v = ln(y) + C. Then, substituting v=x^-1 and multiplying by y, you will get yx^-1 = ln(y) + C. From there, you can solve for y and get the correct general answer of y(x+1) = Cx.

As for your question about obtaining integrating factors and changing the form of the equation, I would suggest reviewing the steps for both of these processes and practicing with different examples. It may also be helpful to work through some problems with a tutor or classmate to get a better understanding.

Overall, the key to solving differential equations is practice and understanding the steps involved. Don't be afraid to ask for help and keep trying - you'll get there!