Graphical solution to pure-time differential equations

Recall facts about derivative

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 $\smash{f'(x)}$?


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.


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