# Solving a differential equation

## 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?

Last edited by a moderator:

Related Calculus and Beyond Homework Help News on Phys.org
ShayanJ
Gold Member
$\partial_y M=\partial_y(xy+y^2+x^2)=x+2y$
$\partial_x N=\partial_x(-x^2)=-2x$
$\partial_y M \neq \partial_x N!!!$

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.

ShayanJ
Gold Member
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.

ShayanJ
Gold Member
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? :/

ShayanJ
Gold Member
$M dx+N dy=0 \Rightarrow N dy=-M dx \Rightarrow \frac{dy}{dx}=- \frac{M}{N}$

Ray Vickson
Homework Helper
Dearly Missed

## 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.

Last edited by a moderator:
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.