### Graphical solution to pure-time differential equations

Sign of derivativeBehavior of function
Derivative positiveFunction increasing
Derivative negativeFunction decreasing
Derivative zeroFunction constant
 How do we do this in general?How to determine $\smash{f(x)}$ from graph of $\smash{f'(x)}$? $f'(x)$ $f(x)$

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 $t \uparrow$ , $y'(t) \to 0$.$\Rightarrow$ graph of $y(t)$ flattens. $y'(t)$ $y(t)$

$y'(6)=0 \Rightarrow$ 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.)