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

Links & Resources

Return to Differential Equations Page
Return to Mini-Course Main Page

Exercises

Directions: You should attempt to solve the problems first and then watch the video to see the solution. 

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1. Solve the differential equation \(y'-2ty=t.\)
   Show Solution

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

   Watch Video Solution

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

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

   Watch Video Solution

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.

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1. Can you define an ODE?

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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?

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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?

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4. Can we solve the ODE \(y'=ty+1\) using the integrating factor method? What about the ODE \(y'=ty^2+1\)? Why?