Use in Solving First Order Linear Ordinary Differential Equations
Integrating factors are useful for solving ordinary differential equations that can be expressed in the form
The basic idea is to find some function, called the "integrating factor," which we can multiply through our DE in order to bring the left-hand side under a common derivative. For the canonical first-order, linear differential equation shown above, our integrating factor is chosen to be
We see that multiplying through by gives
By applying the product rule in reverse, we see that the left-hand side can be expressed as a single derivative in
We use this fact to simplify our expression to
We then integrate both sides with respect to, obtaining
Finally, we can move the exponential to the right-hand side to find a general solution to our ODE:
In the case of a homogeneous differential equation, in which, we find that
where is a constant.
Read more about this topic: Integrating Factor
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