# 8.3: Charged Up

**At Grade**Created by: CK-12

*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, \begin{align*}q\end{align*}

The first step is to separate the variables, and then integrate and solve for \begin{align*}y\end{align*}

2. Find \begin{align*}y(0)\end{align*}

Let’s return to the capacitor. Now that it is discharged, we need to get it charged up again. \begin{align*}A\ 9V\end{align*}

The conservation of energy gives us the differential equation \begin{align*}\frac{dq}{dt} \cdot R = V - \frac{q}{C} \rightarrow \frac{dq}{dt} \cdot R \cdot C = V \cdot C - q\end{align*}

After substituting the given information and simplifying, we get the differential equation \begin{align*}10 \frac{dq}{dt} = 0.9 - q\end{align*}

3. For the differential equation \begin{align*}10 \frac{dq}{dt} = 0.9 - q\end{align*}

4. Apply the initial condition when \begin{align*}\text{time} = 0\end{align*}

The syntax for **deSolve** is **deSolve**\begin{align*}(y'=f(x,y),x,y)\end{align*}**deSolve** command can be found in the **HOME** screen by pressing \begin{align*}\Box\end{align*}**C:deSolve(**.

5. On the **HOME** screen, type **deSolve(**\begin{align*}10q'=0.9-q\end{align*}**and** \begin{align*}q(0)=0,t,q\end{align*}**)**. Write down this answer and reconcile it with your previous solution.

6. In the **HOME** screen enter **deSolve**\begin{align*}(y'=y/x,x,y)\end{align*}

## 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 \begin{align*}C\end{align*}**@**** 7**). Show all the steps by hand if your teacher instructs you to do so.

1. \begin{align*}y' = k \cdot y\end{align*}

2. \begin{align*}y' = \frac{x}{y}\end{align*}

3. \begin{align*}y' = \frac{2x}{y^2}\end{align*}

4. \begin{align*}y' = \frac{3x^2}{y}\end{align*}

Open the **GDB** graph labeled *diffq1*. Observe the family of solutions to the differential equation from Question 4, \begin{align*}y' = \frac{3t^2}{y}\end{align*}

5. Not all differential equations are separable. Use **deSolve** to find the solution to the non-separable differential equation \begin{align*}x \cdot y' = 3x^2 + 2 - y\end{align*}*diffq2* to view graph.

Find the particular solution for the following equations. Show your work. Solve for \begin{align*}y\end{align*}

6. \begin{align*}y' = x \cdot y^2\end{align*}

7. \begin{align*}y' = 1+ y^2\end{align*}

8. \begin{align*}y' = 7y\end{align*}