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Non-separable first order differential equations

dayflower2016

Revising for a maths resit, simply cannot understand how you solve a non-separable first order differential equation by substitution. Have gone through forums, wikipedia, relevant chapters in my textbook and youtube tutorials and everyone seems to do something different, I can't translate it to any questions I attempt.
If someone could explain how you're supposed to do this in teeny tiny baby steps I'll be very grateful, it's doing my nut in!

Reply 1

B_9710

So maybe starting with a simple example? Say you have

\displaystyle x\frac{dy}{dx} =y

, would you know where to start?

Reply 2

dayflower2016

Original post by B_9710

So maybe starting with a simple example? Say you have

\displaystyle x\frac{dy}{dx} =y

, would you know where to start?

I can do that one, because it's separable, could you take me through something like

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

please? Very grateful for help!

Reply 3

B_9710

Oh non separable. Apologies.
So we have

\displaystyle 2xy \frac{dy}{dx}=-(x^2+y^2)

.
I'll be back in a few mins.

Reply 4

NeverLucky

Are you given a substitution for that question? Usually you're given a substitution to use for a differential equation using substitution.

Reply 5

math42

Original post by dayflower2016

I can do that one, because it's separable, could you take me through something like

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

please? Very grateful for help!

Having a brief look, you can rearrange to make dy/dx the subject and then a substitution should "jump out", the following workings are maybe slightly tricky but should lead you to an answer. Some restrictions should probably be on x and y in this tbh but oh well

Reply 6

B_9710

This is a Bernoulli type.
If you rearrange it to give

\displaystyle \frac{dy}{dx} = -\frac{y}{x} - \frac{x}{y}

so you should be able to see it is form the form

\displaystyle \frac{dy}{dx} = p(x)y+q(x)y^n

.
So we multiply through by

y

and make the substitution

v=y^{2}

. Then you can use an integrating factor.

Reply 7

dayflower2016

Original post by NeverLucky

Are you given a substitution for that question? Usually you're given a substitution to use for a differential equation using substitution.

No substitution, I'm supposed to know how to work out what the substitution should be (which I don't). Also there are initial conditions given that y0=1 and x0=0