# Polar plot

The polar coordinates $r$ and $\theta$ can be converted to the Cartesian coordinates $x$ and $y$ using the trigonometry functions: $\begin{cases} x=r\cdot\cos(\theta) \\ y=r\cdot\sin(\theta)\end{cases}$

It follows that a figure specified in $(r, \theta)$ can be plotted by ‘plot2d’ as a parametric curve:

Fig. 1 $r = \cos(5\theta)$

It is possible to plot two or more parametric curves together:

Fig. 2 $r=\theta$ and $r=\cos(5\theta)$

An alternate is the ‘draw2d’ function, it draws graphic objects created by the ‘polar’ function:

Fig. 3 $r=\theta$ and $r=\cos(5\theta)$

Fig. 4 shows a graceful geometric curve that resembles a butterfly. Its equation is expressed in polar coordinates by $r = e^{\sin(\theta)} - 2\cos(4\theta)+\sin(\frac{\theta}{12})^5$

Fig. 4

# Let’s combine them!

It is possible to combine two or more plots into one picture.

For example, we solve the following initial-value problem $\begin{cases} \frac{dy}{dx} = (x-y)/2 \\ y(0)=1 \end{cases}\quad\quad\quad(\star)$

and plot the analytic solution in Fig. 1.

Fig. 1

We can also solve $(\star)$ numerically and plot the discrete data points:

Fig. 2

Fig. 3 is the result of combining Fig.1 and Fig. 2.

Fig.3

It validates the numerical solution obtained by ‘rk’: the two figures clearly overlapped.

# An Alternate Solver of ODEs

Besides ‘ode2’, ‘contrib_ode’ also solves differential equations.

For example, $\frac{d^2y}{dx^2} - \frac{1+x}{x} \cdot \frac{dy}{dx}+ \frac{1}{x}\cdot y=0.$

While ‘ode2’ fails:

‘contrib_ode’ succeeds:

This is an example taking from page 4 of Bender and Orszag’s “Advanced Mathematical Methods for Scientists and Engineers“. On the same page, there is another good example: $\frac{dy}{dx} = \frac{A^2}{x^4}-y^2, \quad\quad A$ is a constant.

Using ‘contrib_ode’, we have

It seems that ‘contrib_ode’ is a better differential equation solver than ‘ode2’:

Even though it is not perfect:

From the examples, we see the usage of ‘contrib_ode’ is the same as ‘ode2’. However, unlike ‘ode2’, ‘contrib_ode’ always return a list of solution(s). It means to solve either initial-value or boundary-value problem, the solution of the differential equation is often lifted out of this list first:

Exercise Solve the following differential equations without using a CAS:

1. $\frac{dy}{dx}= \frac{A^2}{x^4} - y^2$ (hint: Riccati Equation)
2. $\frac{d^2 y}{dx^2} = \frac{y \frac{dy}{dx}} {x}$

# DIY

As far as I know, ‘bc2’, Maxima’s built-in function for solving two-point boundary value problem only handles the type: $\begin{cases} y'' = f(x, y, y')\\ y(a)=\alpha, y'(b) = \beta \end{cases}\quad\quad\quad(*)$

For example, solving $\begin{cases} y'' + y (y')^3=0 \\y(0) = 1, y(1)=3 \end{cases}:$

But ‘bc2’ can not be applied to $\begin{cases} -y'' + 9 y =12 \sin(t)\\ y(0)=y(2\pi), y'(0) = y'(2\pi) \end{cases}$

since it is not the type of (*). However, you can roll your own:

An error occurs on the line where the second boundary condition is specified. It attempts to differentiate the solution with respect to $t$ under the context that $t=0$. i.e.,

diff(rhs(sol), 0);

which is absurd.

The correct way is to express the boundary conditions using ‘at’ instead of ‘ev’:

The following works too as the derivative is obtained before using ‘ev’:

Nonetheless, I still think using ’at’ is a better way: