Integrating Factor - Use in Solving First Order Linear Ordinary Differential Equations

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.


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