Differential equations are tested every year. The actual solving of the differential equation is usually the main part of the problem, but it is accompanied by a related question such as a slope field or a tangent line approximation. BC students may also be asked to approximate using Euler’s Method. Large parts of the BC questions are often suitable for AB students and contribute to the AB sub-score of the BC exam.
What students should be able to do
- Find the general solution of a differential equation using the method of separation of variables (this is the only method tested).
- Find a particular solution using the initial condition to evaluate the constant of integration – initial value problem (IVP).
- NEW Determine the domain restrictions on the solution of a differential equation. See this post for more on this.
- Understand that proposed solution of a differential equation is a function (not a number) and if it and its derivative are substituted into the given differential equation the resulting equation is true. This may be part of doing the problem even if solving the differential equation is not required (see 2002 BC 5 – parts a, b and d are suitable for AB)
- Growth-decay problems.
- Draw a slope field by hand.
- Sketch a particular solution on a (given) slope field.
- Interpret a slope field.
- Use the given derivative to analyze a function such as finding extreme values
- For BC only: Use Euler’s Method to approximate a solution.
- For BC only: use the method of partial fractions to find the antiderivative after separating the variables.
- For BC only: understand the logistic growth model, its asymptotes, meaning, etc. The exams so far, have never asked students to actually solve a logistic equation IVP
Look at the scoring standards to learn how the solution of the differential equation is scored, and therefore, how students should present their answer. This is usually the one free-response answer with the most points riding on it. Starting in 2016 the scoring has changed slightly. The five points are now distributed this way:
- one point for separating the variables
- one point each for finding the antiderivatives
- one point for including the constant of integration and using the initial condition – that is, for writing “+ C” on the paper with one of the antiderivatives and substituting the initial condition; finding is value is included in the “answer point.” and
- one point, the “answer point”, for the correct answer. This point includes all the algebra and arithmetic in the problem.
A future post suggested by a reader and tentatively schooled for April 4, 2017, will discuss the domain of the solution of a differential equation. The domain is often included on the scoring standard, but unless it is specifically asked for in the question students do not need to include it. This is a new topic not previously on the Course Description.
Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.
Revised 3-21-17, 08:21 to reflect new point distribution.
Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)
Tuesday March 28: for BC Parametric Equation (Type 8)
Friday March 31: For BC Polar Equations (Type 9)
Tuesday April 4: For BC Sequences and Series.
Friday April 7, 2017 The Domain of the solution of a differential equation.