# Section 4: Variation of Parameters

### 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.
• When you have finished the material below, you can start on the next section or return to the main differential equations page.

### Concepts

• Solving first-order linear equations using the method of variation of parameters

### Exercises

Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. Solve the following differential equation using variation of parameters $ty'+y=t\cos(2t), \quad t>0$

$$y=\dfrac{1}{2}\sin(2t)+\dfrac{1}{4t}\cos(2t)+\dfrac{C}{t}$$

### 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. How many independent variables are there in any system of ODEs?
2. How many initial conditions should be coupled with a system of ODEs to form an IVP?
3. Give an example of a linear, nonhomogeneous and non-autonomous system of ODEs. Then write it in matrix form.