Differential Equations and Integrating Factors

by Joe Archive on June 23, 2011

For “fun” over the last few days, I’ve been looking at some of my old undergraduate textbooks and trying to see what I remembered. One thing I was vaguely still aware of (from 1996 or so) was the use of integrating factors to solve simple first order ODEs. So, once I figured them out again, I thought I’d write it all down here to help me remember for next time.

So, imagine you have an ODE like this:

yprime + p(x) y = g(x).

Now, it would be nice to simplify the left hand side – to fo this, consider the following:

partial_x e^{left(int p(x)dxright)} y = e^{left(int p(x)dxright)} partial_xy + p(x)e^{left(int p(x)dxright)} y,

which is the left hand side, multiplied by the integrating factor e^{left(int p(x)dxright)}. So, if we multiply our original ODE by the integrating factor, we can then make the simplification:

e^{left(int p(x)dxright)}(y'+p(x)y)=e^{left(int p(x)dxright)}g(x) partial_xleft(e^{left(int p(x)dxright)}y right)=e^{left(int p(x)dxright)} g(x) , e^{left(int p(x)dxright)}y = int e^{left(int p(x)dxright)} g(x) dx +c, y = frac{int e^{left(int p(x)dxright)} g(x) dx +c}{e^{left(int p(x)dxright)}},

 

where c is the arbitrary constant from the integration of the right hand side, and can often be evaluated with initial (or boundary) conditions.

 

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