8.3: Charged Up
This activity is intended to supplement Calculus, Chapter 7, Lesson 7.
ID: 12368
Time Required: 20 minutes
Activity Overview
In this activity, students will be asked some exam-like questions to introduce separable differential equations. Students will use the deSolve command to find general and particular solutions to differential equations. They will also graphically view the family of particular solutions to a differential equation.
Topic: Differential Equations
- Differential equations solved algebraically. Solutions shown graphically.
- Use TI-89 to find solutions to DEs.
Teacher Preparation and Notes
- The syntax for deSolve is deSolve\begin{align*}(y'=f(x,y),x,y)\end{align*} where \begin{align*}x\end{align*} is the independent and \begin{align*}y\end{align*} is the dependent variable. The deSolve command can be found in the HOME screen by pressing \begin{align*}F3\end{align*} or \begin{align*}F4\end{align*} and selecting C: deSolve(. This activity will help students see the application of DEs, the process of solving separable DEs, and how to use the TI-89 to check or find the solution to DEs.
- Students will write their responses directly on the accompanying handout.
- To download the calculator file, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=12368 and select Diffq1 and Diffq2.
Associated Materials
- Student Worksheet: Charged Up http://www.ck12.org/flexr/chapter/9732, scroll down to the third activity.
- Diffq1.89d
- Diffq2.89i
Part 1 – Separable Differential Equation Introduced
For Questions 1 and 2, students are asked exam like questions about differential equations. These foundational (practically review questions) can serve as an introduction to separable differential equations. Teachers can use question 2 to formatively assess how well students are making the connection to their previous knowledge. If students appear to be struggling, teachers can use the opportunity to state the following steps: STEP 1 separate variable, STEP 2 integrate both sides, STEP 3 apply initial conditions to solve for the constant of integration and find a particular solution, STEP 4 answer the question.
DEs are applicable for all sorts of physical phenomena, including radioactive decay, economics, biology, chemistry, population growth and electric circuits with capacitors.
The steps for solving a separable DE are outlined and reinforced. Some students may need help with the algebra involved in the steps.
The deSolve command is used to check the solution. The syntax of deSolve is explained so that students can use this as a tool to explore several other DEs.
Student Solutions
1. \begin{align*} \frac{dq}{dt} = kq\end{align*}
2. \begin{align*}\int \limits dy = \int \limits \sin (x) \cos^2(x) dx \Rightarrow y = - \frac{1}{3} \cos^3(x) + C\end{align*} and since \begin{align*}\cos \left( \frac{\pi}{2} \right) = 0, y(0) = - \frac{1}{3}\end{align*}.
3. \begin{align*}\frac{dq}{0.9 - q} = \frac{1}{10} dt \Rightarrow - \ln |0.9 - q| = \frac{1}{10} t + C \Rightarrow \ln |0.9 - q| = - \frac{1}{10} t + C\end{align*}
4. \begin{align*}|0.9 - q| = e^{- \frac{1}{10} t + C} = e^{- \frac{1}{10} t} e^C\end{align*}. Let \begin{align*}C_1 = \pm e^C, 0.9 - q = C_1 e^{- \frac{1}{10} t}\end{align*} and with \begin{align*}q = 0\end{align*} when \begin{align*}t = 0, C_1 = 0.9\end{align*}
\begin{align*}q(t) = 0.9 \left(1- e^{-\frac{1}{10}t} \right)\end{align*}
5. They are equivalent. In the solution using deSolve, the 0.9 was distributed
6. \begin{align*}y = c \cdot x\end{align*}
If \begin{align*}y(1) = 1, c = 1\end{align*}, so the particular solution is \begin{align*}y = x\end{align*}.
Part 2 – Homework/Extension – Practice with deSolve and Exploring DEs
Students first use deSolve to find the general solution to four DEs. The fourth situation can also be viewed graphically. To view the graph of the family of functions, students should enter RclGDB diffq1 on the HOME screen
On the worksheet students will write the solution to each DE. You may wish for students to algebraically show their work and use deSolve check their answers.
A non-separable DE is solved with deSolve. The particular solution can also be viewed graphically by entering RclPic diffq2 on the HOME screen. The slope field and the particular solution are shown on the screen.
Finally, deSolve is used to find the particular solution for three DEs with initial conditions. Students are asked to solve for \begin{align*}y\end{align*}. With the TI-89, this can be done with solve(). Again, students may show their work and use deSolve to check their answers.
Another extension/exploration activity would be to have students come up with their own DE and find the general solution. Have them discuss solutions that surprised them.
Student Solutions
- \begin{align*}y = Ce^{kx}\end{align*}
- \begin{align*}y^2 = x^2 + C\end{align*}
- \begin{align*}y^3 = 3x^2 + C \to y = (3x^2 + C)^{\frac{1}{3}}\end{align*}
- \begin{align*}y^2 = 2x^3 + C\end{align*}
- It looks like a parabola because \begin{align*}y = x^2 + \frac{c}{x} + 2\end{align*} is \begin{align*}y = x^2 + 2\end{align*} when \begin{align*}c = 0\end{align*}.
- \begin{align*}1 - \frac{1}{y} = \frac{x^2}{2} \Rightarrow y = \frac{-2}{x^2 - 2}\end{align*}
- \begin{align*}\tan^{-1}(y) + \frac{\pi}{4} = x \to y = \tan \left(x - \frac{\pi}{4} \right)\end{align*}
- \begin{align*}y = e^{7x}\end{align*}