Throughout the chapter, we worked through several examples, each demonstrating how to apply the integrating factor method in different contexts. In each case, the systematic process of identifying the standard form, computing the integrating factor, and applying integration was key to solving the differential equation.
Summary of the Key Ideas.
First-Order Linear Differential Equations
First-order linear differential equations take the standard form
The integrating factor method is the process of solving a first order linear differential equation by turning it into a direct integration problem.
ExercisesExercises
Forward Product Rule.
1.
\(\, \ds f(x) = \ln x\cos x \)
Answer.
\(\ds
f'(x) = \left(\frac{1}{x}\right)\cos x + \ln x \left(-\sin x\right)
= \frac{\cos x}{x} - \ln x \sin x\)
The Integrating Factor.
2.
\(\ds y' - 4y = x \)
Answer.
\(\ds \)
Conceptual Questions.
3.
Write the differential equation below in the form \(y'+p(x)y=q(x)\) and identify \(p(x)\) and \(q(x)\text{.}\) Also, state the order and whether it is linear or not.