# Section 1: Integrating Factor

### Instructions

• First, you should watch the concepts videos below explaining the topics in the section.
• Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
• Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
• When you have finished the material below, you can start on the next section or return to the main differential equations page.

### Concepts

• Definitions and terminology for ordinary differential equations
• Using an integrating factor to solve a first-order linear differential equation

### Exercises

Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. Solve the differential equation $$y'-2ty=t.$$

$$y=-\dfrac{1}{2}+Ce^{t^2}$$

2. Solve the initial value problem below and find the interval of validity of the solution$(\cos x)y'+(\sin x) y =1, \quad y(\pi)=2$

$$y=\sin x-2\cos x$$, Interval of Validity: $$(-\infty, \infty)$$

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. Can you define an ODE?
2. Give an example of a $$3^{\mbox{rd}}$$ order ODE, with $$t$$ as your independent variable and $$x=x(t)$$ as the dependent variable. Is your example an autonomous ODE?
3. What is the significance of an integrating factor? How does it lead to finding the solutions of a linear, $$1^{\mbox{st}}$$ order, ODE?
4. Can we solve the ODE $$y'=ty+1$$ using the integrating factor method? What about the ODE $$y'=ty^2+1$$? Why?