Chapter 2

Visualizing a certain first-order autonomous ODE.

Visualizing $Y’ = AY + B$.

Here we visualize the solution $Y(t)$ to the autonomous ODE

\begin{align} Y' +AY = B, \end{align}

with constants $A ,B$ ($A \ne 0$) and imposed initial condition $Y(0) = Y_0$.

As solved in the textbook, the solution to this IVP is given by the formula $$Y(t) = (Y_0 - \frac{B}{A}) e^{-At} + \frac{B}{A}.$$

To see the visualization, choose values for the constants $A$, $B$ and choose the intial condition $Y(0)=Y_0$ at time $t=0$. When you keep $A$ fixed, observe the curve changes monotonicity when $Y_0 - \dfrac{B}{A}$ changes sign. When you keep $B,Y_0$ fixed and change $A$, observe that the curve changes monotonicity only when exactly one of $A$ and $Y_0 - \dfrac{B}{A}$ changes sign.

Constants and Initial Condtions

$A$:

$B$:

$Y_0$:

$A > 0,$ $A < 0,$

$Y_0 - \dfrac{B}{A} = $ $> 0$.$= 0$.$< 0$.

Graph