OBEYDUR RAHMAN
Aug 16, 2009, 01:50 AM
Hi
Would you help me to tell about integrating factor
galactus
Aug 16, 2009, 05:19 AM
Post a specific problem and we can show you.
Also, you can Google it and find plenty.
But here is an explanation:
Sometimes we can't separate variables, so we can use the integrating factor that Euler developed.
A first-order DE is called linear if we can write it as \frac{dy}{dx}+p(x)y=q(x)... [1]
Where p(x) and q(x) may be constants.
We can make the observation that if we define {\mu}={\mu}(x) by {\mu}=e^{\int p(x)dx}, then
\frac{d{\mu}}{dx}=e^{\int p(x)dx} \cdot \frac{d}{dx}\int p(x)dx={\mu}p(x)
Therefore:
\frac{d}{dx}[{\mu}y]={\mu}\frac{dy}{dx}+\frac{d{\mu}}{dx}y={\mu}\frac{ dy}{dx}+{\mu}p(x)y... [2]
If we multiply [1] through by {\mu}, and then simplify using [2], we get:
{\mu}\frac{dy}{dx}+{\mu}p(x)y={\mu}q(x)
\frac{d}{dx}({\mu}y)={\mu}q(x)
This can be solved by integrating both sides to get:
{\mu}y=\int {\mu}q(x)dx+C \;\ or \;\ y=\frac{1}{\mu}\left[\int {\mu}q(x)dx+C\right]
There ya' go. A general explanation. If you want a specific example, please post a problem you are having trouble with and I can try to guide you through it. Okey-doke?
It's not as bad as it looks.