Recall facts about derivative

Sign of derivative | Behavior of function |
---|---|

Derivative positive | Function increasing |

Derivative negative | Function decreasing |

Derivative zero | Function constant |

How do we do this in general? | $f'(x)$ | $f(x)$ |

Use derivative facts to sketch solution of pure-time differential equation

\begin{align*} y'(t) &= 0.5t-3\\ y(0) &= 4 \end{align*} | (Initial condition) |

Sketch graph of $y'(t)$. When $t=0$, what is sign of $y'(t)$? Negative. $y(t)$ decreases at $t=0$. As $t \uparrow$ , $y'(t) \to 0$. | $y'(t)$ | $y(t)$ |

$y'(6)=0 \Rightarrow$ graph of $y(t)$ is horizontal at $t=6$.

For $t>6$, $y'(t)>0 \Rightarrow$ $y(t)$ increases.

Graph of $y(t)$ represents solution.

Second example: Solve the pure-time differential equation

\[{}\] | where $f(t)$ is plotted below. |

Step 1: Find $t$ where $f(t)=0$.

$f(t)=0$ when $t=1, 3, 6$

The graph of $y(t)$ must be horizontal at these points.

Step 2: Determine sign of $f(t)$ in between.

- $f(t) \lt 0$ for $t \lt 1$
- $f(t) \gt 0$ for $1 \lt t \lt 3$
- $f(t) \lt 0$ for $3 \lt t \lt 6$
- $f(t) \gt 0$ for $t \gt 6$

Step 3: Start at initial condition.

Step 4: Sketch solution that moves in the correct directions.(If change initial condition, graph just shifts up or down.)