Question

Solve xyy'=3x^6+6y^2 explicitly.

Answer 1

@SithsAndGiggles

Answer 2

If I use Bernoulli's equation in this case, what's v?

Answer 3

Another Bernoulii sub I believe. \(v=y^2\), so \(v'=2yy'\).

Answer 4

You want to choose \(v\) so that it's derivative contains at least the same power of \(y\) as the term with \(y'\) in the initial equation. In this case, the \(y'\) term is \(\cdots yy'\), so you want something like \(v=\cdots y^2\). You can always manipulate the constants.

Answer 5

So after I sub, I get xv'/2=3x^6+6v, right?

Answer 6

Right. The equation is linear now.

Answer 7

*almost linear. You just need to do the proper rearrangement.

Answer 8

xv'/2-6v=3x^6?

Answer 9

Yes, and divide everything by \(\dfrac{1}{2}x\) too.

Answer 10

And the integrating factor is e^-6x?

Answer 11

No the IF is something else.
\[\frac{1}{2}xv'-6v=3x^6~~\iff~~v'-\frac{12}{x}v=6x^5\]
with
\[IF=\exp\left(-\int\frac{12}{x}~dx\right)=\cdots\]

Answer 12

So IF=1/x^12?

Answer 13

Yes

Answer 14

Thanks!

Answer 15

yw