Chapter 5

Visualizing second-order linear ODE’s and complex numbers.

Second-order linear ODE’s with constant coefficients.

Here we visualize the solutions to the second-order linear ODE

\begin{align} y'' + py' + qy = 0, \end{align}

with constants $p,q$.

To play with the visulization yourself, pick values for the constants $p,q$, specify the initial conditions $y(0)$ and $y'(0)$, and choose your prefered display options:

Constants

$p$:

$q$:

Initial Condtions

$y(0)$:

$y'(0)$:

Display Options Independent solutions:
hide
show

Outputs

As we have learned, the associated characteristic polynomial is $\lambda^2 + p \lambda + q = \lambda^2$+$\lambda$-1.5. The roots are $\lambda_1 =$ , $\lambda_2 =$ .

Two independent solutions are $y_1(t) = e^{\lambda_1 t}$ and $y_2(t) = e^{\lambda_2 t}$.

Two independent solutions are $y_1(t) = e^{\lambda_1 t}$ and $y_2(t) = te^{\lambda_1 t}$.

Let $\lambda_1, \lambda_2 = r_0 \pm i s_0$. Then two independent solutions are $y_1(t) = e^{r_0t}\cos(s_0t)$ and $y_2(t) = e^{r_0t}\sin(s_0t)$.

$y(t) = c_1y_1(t) + c_2y_2(t) = $ $y_1(t)$$+$ $-$ $y_2(t)$.

Graph

Visualizing complex numbers.

To visualize Euler's formula, move the point $P$ on the circle:

Outputs

$x = \operatorname{cos} \theta = $ , $y= \operatorname{sin} \theta = $ .

$\theta = $ radians $ = $ $\pi$ radians = degrees.

Euler's Formula

$e^{i\theta} = \operatorname{cos} \theta + i \operatorname{sin} \theta = $ $+$$-$ $i$.

Euler's Formula

To visualize complex multiplication, move the points $z_1,z_2$ on the respective circles:

Outputs

$z_1 = 1.5e^{i \theta_1} $$=1.5\operatorname{cos} \theta_1 + (1.5\operatorname{sin} \theta_1)i = $ $ + ($$)i$.

$z_2 = 2e^{i \theta_2} $$=2\operatorname{cos} \theta_2 + (2\operatorname{sin} \theta_2)i = $ $ + ($$)i$.

$z_3 = z_1z_2$$=3e^{i (\theta_1+\theta_2)}$$=3e^{i \theta_3}$$=3\operatorname{cos} \theta_3 + (3\operatorname{sin} \theta_3)i$$=$ $ + ($$)i$.

$\theta_1 = $ radians $=$ $\pi$ radians $=$ degrees.

$\theta_2 = $ radians $=$ $\pi$ radians $=$ degrees.

$\theta_3$$=(\theta_1+\theta_2) \text{ mod } 2\pi = $ radians $=$ $\pi$ radians $=$ degrees.

Complex Multiplication

To visualize complex conjugation and inversion, move the point $z$ on its circle:

Outputs

$z = 2e^{i \theta} $$=2\operatorname{cos} \theta + (2\operatorname{sin} \theta)i = $ $ + ($$)i$.

$\overline{z} = 2e^{-i \theta} $$=2\operatorname{cos} \theta - (2\operatorname{sin} \theta)i = $ $ + ($$)i$.

$1/z = \overline{z}/|z|^2$$=0.5e^{-i \theta}$$=0.5e^{-i \theta}$$=0.5\operatorname{cos} \theta - (0.5\operatorname{sin} \theta)i$$=$ $ + ($$)i$.

$\theta = $ radians $=$ $\pi$ radians $=$ degrees.

Complex Conjugation and Inversion