# 8.3: Charged Up

*This activity is intended to supplement Calculus, Chapter 7, Lesson 7.*

## Part 1 – Separable Differential Equations Introduced

1. A capacitor, like one used for a camera flash, is charged up. When it discharges rapidly the rate of change of charge, , with respect to time, , is directly proportional to the charge. Write this as a differential equation.

The first step is to separate the variables, and then integrate and solve for .

2. Find , if and . After integrating, use the initial condition when to find the constant of integration. Then, substitute to find .

Let’s return to the capacitor. Now that it is discharged, we need to get it charged up again. battery is connected to a resister, , and capacitor, .

The conservation of energy gives us the differential equation .

After substituting the given information and simplifying, we get the differential equation .

3. For the differential equation , separate the variables and integrate.

4. Apply the initial condition when , the charge and solve for .

The syntax for **deSolve** is **deSolve** where is the independent and is the dependent variable. The **deSolve** command can be found in the **HOME** screen by pressing and selecting **C:deSolve(**.

5. On the **HOME** screen, type **deSolve(** **and** **)**. Write down this answer and reconcile it with your previous solution.

6. In the **HOME** screen enter **deSolve** to find the general solution of . Write the answer. Show your work to solve this differential equation by hand and apply the initial condition to find the particular solution.

## Part 2 – Homework/Extension – Practice with deSolve and Exploring DEs

Find the general solution for the following separable differential equations. Write the solution in an acceptable format, (for example, use instead of **@**** 7**). Show all the steps by hand if your teacher instructs you to do so.

1.

2.

3.

4.

Open the **GDB** graph labeled *diffq1*. Observe the family of solutions to the differential equation from Question 4, . Many particular solutions can come from a general solution. When you are finished viewing the family of functions, go to the screen and delete the function.

5. Not all differential equations are separable. Use **deSolve** to find the solution to the non-separable differential equation . What does this graph look like if the integration constant is 0? Explain. Open picture *diffq2* to view graph.

Find the particular solution for the following equations. Show your work. Solve for . Explore other DEs on your own. Do you get any surprising results?

6. and

7. and

8. and