Determine general solutions to differential equations
I can use separation of variables to find a general solution to a differential equation
Quick Lesson Plan
Are you a Solution Seeker? guides students through the prerequisite skills for applying separation of variables. The lesson first reviews using slope fields to predict a family of solution curves. Groups then navigate equations with integral expressions on both sides of the equal sign. Finally, the process and our vocabulary of separation of variables is introduced: Separate, Integrate, Isolate. The majority of class time is devoted to Check Your Understanding. This is when students will have to combine their calculus and algebra skills in order to find success!
For maximum effect, present this lesson as an opportunity for students to complete two integrals for the price of one! But then impress upon them that only one constant of integration is required. Generally, the “plus C” is placed on the right side, but everyone’s final solutions will be equivalent if their algebra is correct.
Algebra concepts to review: Most students need a review of absolute value notation and meaning. And many will finally understand the importance of absolute value in a natural log antiderivative. Powers that have a sum as the exponent should be rewritten as a multiplication expression in order to create coefficients from constant terms (see Activity #3 or CYU #1).
A recent FRQ (2019 AB4) gives students an interesting application of differential equations. The process for solving the given differential equation is straightforward separation of variables. However, the algebra and antiderivative required prove to be challenging for a majority of students. From integrating “a square root on the bottom” to expanding a binomial (generally not required on the AP Test), this problem presents obstacles throughout.
Another interesting application of separation of variables is found in 2006 AB5. The scoring rubric for part b is typical and should be shared with students often --- they need to know the many opportunities they have for earning points on an FRQ problem!
Beware: Students will try to separate variables with illegal maneuvers! For example, using subtraction or addition, or even introducing additional variables into the original differential equation, will prevent them from earning any points on an FRQ. Emphasize that separation can be accomplished only through division or multiplication. Have students work problems that require factoring and/or division by a binomial term (CYU #3).