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Finding Particular Solutions using Initial Conditions and Separation of Variables (Topic 7.7)

Unit 7 - Day 6

​Learning Objectives​
  • Determine particular solutions to differential equations

​Success Criteria
  • I can write a solution to a differential equation using an integral expression and an initial condition

  • I can use the initial condition to solve for “C” to find a particular solution to a differential equation

Quick Lesson Plan
Activity: How many Sea Lions are on Elliott Bay?



Lesson Handout

Answer Key


Topic 7.7 directs students to find one particular solution to a differential equation. This requires the use of an initial condition: information that describes a function value for a specific input. Building on the Important Ideas from Topic 7.6 (Separate, Integrate, Isolate), we add two more components to the sequence: solving for the constant of integration, C, and selecting the proper solution.  If the antiderivative is unknown, using an integral expression as the solution is necessary and is practiced in the Check Your Understanding section.  

Teaching Tips

Using correct algebra may be the most important lesson today!! Applying the inverse functions of logs and exponentials is often challenging, as is resolving absolute value expressions. When using an integral expression as the solution to a differential equation, remind students that the upper limit of integration is the independent variable for the integral function. They should strive to use a different variable in the integrand. 

Remember: A sketch of the particular solution on a slope field must contain (go through) the initial (given) condition. 

Exam Insights

When discussing Check Your Understanding #1ab, encourage students to add the constant value at the front of the integral expression. Writing the constant after the integral will be problematic for students who forget to write a “dx” or “dt” term.  AP Readers will deduct a point if the integrand seems to incorrectly contain the constant value. Adding the constant term in front of the integral will avoid this issue!

Student Misconceptions

Occasionally, require students to find the derivative of their solution to verify their work. To convince students that an integral expression may be the solution to a differential equation, have them use the FTC to differentiate their integral. They should see the original “DiffEq” reappear. 

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