# Thread: Problem with Integrating Factor

1. ## Problem with Integrating Factor

I'm trying to solve the differential equation dy/dx + (2-3x^2)(x^-3)y = 1. I've integrated (2-3x^2)(x^-3) and taken the exponential of this to give an integrating factor of (x^-3)e^(x^2). However, when I differentiate the integrating factor multiplied by y, I don't end up with the left hand side of the equation. I must have gone wrong somewhere in my working, however, I've checked it several times!

2. Originally Posted by StaryNight
I'm trying to solve the differential equation dy/dx + (2-3x^2)(x^-3)y = 1. I've integrated (2-3x^2)(x^-3) and taken the exponential of this to give an integrating factor of (x^-3)e^(x^2). However, when I differentiate the integrating factor multiplied by y, I don't end up with the left hand side of the equation. I must have gone wrong somewhere in my working, however, I've checked it several times!
I think you'll find that in the integrating factor it's $e^{-\frac{1}{x^2}$ not $e^{x^2}$. Note: $-\frac{1}{x^2} \neq x^2$ .... it seems you're confusing $-\frac{1}{x^2}$ with $\frac{1}{x^{-2}}$ ....

3. Originally Posted by StaryNight
I'm trying to solve the differential equation dy/dx + (2-3x^2)(x^-3)y = 1. I've integrated (2-3x^2)(x^-3) and taken the exponential of this to give an integrating factor of (x^-3)e^(x^2). However, when I differentiate the integrating factor multiplied by y, I don't end up with the left hand side of the equation. I must have gone wrong somewhere in my working, however, I've checked it several times!
$\displaystyle \int \frac{2-3x^2}{x^3}=-\frac{1}{x^2}-3lnx+C$

Integrating Factor = $\displaystyle e^{-\frac{1}{x^2}-3lnx}=e^{-\frac{1}{x^2}}e^{-3lnx}=x^{-3}e^{-\frac{1}{x^2}}$

4. We can rewrite the equation in the form:

$Pdx+Qdy=0\;,\quad P=(1-3x^2)y\;,\;Q=x^3$

and:

$\dfrac{1}{Q}\left(\dfrac{{\partial P}}{{\partial y}}-\dfrac{{\partial Q}}{{\partial x}}\right)=\ldots=\dfrac{1}{x^3}-\dfrac{6}{x}=F(x)$

so, an integrating factor is:

$\mu(x)=e^{\int F(x)\;dx}$

Fernando Revilla

Edited: Sorry, I misread the OP's equation.