Integrating Factors: A Shortcut

Shortcuts are dangerous things – they may save you time, but they usually don’t help you understand the problem. ¬†Because of this, it’s usually important to have a thorough grasp of the basic idea of how to solve a problem before learning the shortcut. ¬†Since you’ve had a week or so wrestle with the “Integrating Factors” problem, I wanted to share a standard shortcut (covered in the text, but not yet discussed in class) for solving these problems, which condenses much of the algebra into two formulas. ¬†You are welcome to use it, or not, as you prefer.

Shortcut for solving Integrating Factors problems:

Step 1:  Rewrite the differential equation in the standard form:  

\frac{dy}{dt} + p(t)y = g(t)

In practice, this usually just means getting the y and \frac{dy}{dt} on the same side, and dividing to get rid of anything in front of the \frac{dy}{dt}.

Step 2:  Find \mu, by plugging in:

\mu = e^{\int p(t) dt}

That is, integrate the function in front of y, and then raise e to the power of the result.  This gives \mu

Step 3:  Find y, by plugging in:

y = \frac{1}{\mu(t)} \int \mu(t) g(t) dt + C

That is, multiply \mu by the function g(t) from the right hand side of the differential equation, integrate, and multiply the result by $\frac{1}{\mu}$.

NOTE: The¬†standard form mentioned in Step 1 shows up a lot – in fact, even if you are not using the shortcut formulas above, it is considered “pretty standard” to rewrite your equation in standard form before solving the problem.

Happy shortcutting,
-Prof Reitz

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