# Solving a differential equation

• MarcL
Thanks for catching that!In summary, the student attempted to solve a homework equation and was stuck because the answer key factors out x^2 from the first equation. He was able to substitute ##y = x u## and solve for ##u## which yielded a simple DE for ##u##.f

## Homework Statement

Solve (xy+y2+x2) dx -( x2 )dy = 0

## The Attempt at a Solution

So I can see it isn't separable and linear, so I thought of solving it through substitution

i did y=ux and dy= u dx + x du
and I substituted them in my first equation giving me
(xu+u2+x^2) dx - x2(udx + xdu) = 0]

I'm stuck here... the answer key factors out x^2 from the first equation leaving it with x^2(u+u^2+1), which doesn't make sense to me.. because if I factored out x^2 I would be left with (x-1u+x-2u2+1)
So any help on how I am seeing this wrong?

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## \partial_y M=\partial_y(xy+y^2+x^2)=x+2y ##
## \partial_x N=\partial_x(-x^2)=-2x ##
## \partial_y M \neq \partial_x N! ##

I know how to verify if exact, hence why I chose substitution, but I am kinda stuck at how to factor my xα. Unless I didn't understand your answer correctly, to me it seems as if you're solving to see whether or not if exact.

That means your differential isn't exact which means it can't be integrated in this form, so you should turn it into an exact differential. There is such a method in your toolbox which you can't integrate the differential without it. You remember it?

Well I know I can use substitution to reach a separable equation. If not, I know I can use υ(x,y) and multiply it to my equation. My question was more towards the substitution method because I don't completely grasp the subject.

By some manipulation, you can get ## y'=\frac{y}{x}+\frac{y^2}{x^2}+1 ##. Now try your substitution!

I'm sorry, I'm not understanding. why you took the derivative of y. I was taught ( and my book explains) a different way. For instance, we just have to find a coeffiecient of xα to then create a separable DE. anyway, I'll check again later, maybe this will help getting a clearer answer? :/

## M dx+N dy=0 \Rightarrow N dy=-M dx \Rightarrow \frac{dy}{dx}=- \frac{M}{N}##

## Homework Statement

Solve (xy+y2+x2) dx -( x2 )dy = 0

## The Attempt at a Solution

So I can see it isn't separable and linear, so I thought of solving it through substitution

i did y=ux and dy= u dx + x du
and I substituted them in my first equation giving me
(xu+u2+x^2) dx - x2(udx + xdu) = 0]

I'm stuck here... the answer key factors out x^2 from the first equation leaving it with x^2(u+u^2+1), which doesn't make sense to me.. because if I factored out x^2 I would be left with (x-1u+x-2u2+1)
So any help on how I am seeing this wrong?

If you substitute ##y = x u## into the DE
$$x^2 \frac{dy}{dx} = x^2 + xy + y^2 \; \Rightarrow \frac{dy}{dx} = 1 + \frac{y}{x} + \left(\frac{y}{x}\right)^2$$
the resulting DE for ##u## is very simple and is straightforward to solve.

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Yeah it was a stupid mistake, I was going fast and stressing out over an exam. I should've noticed that replacing y = ux into the equation can allow me to factor out x^2 because the function is homogeneous.