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 $t=6$.
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.
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.)