Question

How does one solve the equation

dy/dx - 4xy = 0

using an integrating factor? I see how to do it using a separation of variables approach, but how can an IF method work when the equation is set to 0?

Answer 1

\[\frac{dy}{dx}-4xy=0\]

Answer 2

Integrating factor is
\[\mu(x)=e^{\int\limits P(x)dx}\]
Where
\[\frac{dy}{dx}+P(x)y=Q(x)\]

Answer 3

But Q(x) is zero, so you end up with y=0

Answer 4

Actually you don't need to, this is separable

Answer 5

I know, but I have to do it using an integrating factor.

Answer 6

Just do it,
\[\mu(x)=e^{\int\limits-4xdx}=e^{-2x^2}\]
Multiply across,
\[e^{-2x^2}\frac{dy}{dx}-4e^{-2x^2}xy=0\]
Then reverse product
\[\frac{d}{dx}(e^{-2x^2}y)=0 \]
Integrate both sides

Answer 7

\[e^{-2x^2}y=c_1 \]
\[y=c_1e^{2x^2}\]

Answer 8

ah ok, that makes sense. cheers dude.

Answer 9

yw

Answer 10

@ParoxysmX ...proceed d same way and you will get it with a little bit rearrangement...
or tell me how you did it and i will try to help..