B3D Handout 12: The Integrating Factor Method.
This method is used to solve equations which are
•Ordinary differential equations
•Linear
•First-order
with any kind of coefficients.
A first order linear differential equation for y(x) must be of the form
dy
dx+p(x)y=q(x).
If there is something multiplying the dy/dxterm, then divide the whole equation by this first.
Now suppose we calculate an integrating factor
I(x) = exp µZp(x) dx¶.
Just this once, we won’t bother about a constant of integration.
We multiply our equation by the integrating factor:
I(x)dy
dx+I(x)p(x)y=I(x)q(x).
and then observe that
d
dx(yI(x)) = dy
dxI(x) + ydI
dx=dy
dxI(x) + yp(x)I(x)
which is our left-hand-side. So we have the equation
d
dx(yI(x)) = I(x)q(x)
which we can integrate (we hope):
yI(x) = ZI(x)q(x) dx+C
y=1
I(x)ZI(x)q(x) dx+C
I(x).
This is the general solution of the equation.
If we have initial conditions, then the very last thing we do is determine the constant C.
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