Chapter 1
Introduction

This gives detailed description of all supported differential equations in my ode solver. Whenever possible, each ode type algorithm is described using flow chart.

Each ode type is given an internal code name. This internal name is used by the solver to determine which specific solver to call to solve the ode.

A differential equation is classified as one of the following types.

1. First order ode.
2. Second and higher order ode.

For first order ode, the following are the main classifications used.

1. First order ode \(f(x,y,y')=0\) which is linear in \(y'(x)\).

2. First order ode not linear in \(y'(x)\) (such as d’Alembert, Clairaut). But it is important to note that in this case the ode is nonlinear in \(y'\) when written in the form \(y=g(x,y')\). For an example, lets look at this ode

   \[ y' = -\frac {x}{2}-1+\frac {\sqrt {x^{2}+4 x +4 y}}{2} \]
   Which is linear in \(y'\) as it stands. But in d’Alembert, Clairaut we always look at the ode in the form \(y=g(x,y')\). Hence, if we solve for \(y\) first, the above ode now becomes
   \begin{align*} y &= x y' + \left ( (y')^{2}+ 2 y' + 1 \right )\\ &= g(x,y') \end{align*}

   Now we see that \(g(x,y')\) is nonlinear in \(y'\). The above ode happens to be of type Clairaut.

For second order and higher order ode’s, further classification is

1. Linear ode.
2. non-linear ode.

Another classification for second order and higher order ode’s is

1. Constant coefficients ode.
2. Varying coefficients ode

Another classification for second order and higher order ode’s is

1. Homogeneous ode. (the right side is zero).
2. Non-homogeneous ode. (the right side is not zero).

All of the above can be combined to give this classification

1. First order ode.

   1. First order ode linear in \(y'(x)\).
   2. First order ode not linear in \(y'(x)\) (such as d’Alembert, Clairaut).

2. Second and higher order ode

   1. Linear second order ode.

      1. Linear homogeneous ode. (the right side is zero).
      2. Linear homogeneous and constant coefficients ode.
      3. Linear homogeneous and non-constant coefficients ode.
      4. Linear non-homogeneous ode. (the right side is not zero).
      5. Linear non-homogeneous and constant coefficients ode.
      6. Linear non-homogeneous and non-constant coefficients ode.

   2. Nonlinear second order ode.

      1. Nonlinear homogeneous ode.
      2. Nonlinear non-homogeneous ode.

For system of differential equation the following classification is used.

1. System of first order odes.

   1. Linear system of odes.
   2. non-linear system of odes.

2. System of second order odes.

   1. Linear system of odes.
   2. non-linear system of odes.

Currently the program does not support Nonlinear higher order ode. It also does not support nonlinear system of first order odes and does not support system of second order odes.

The following is the top level chart of supported solvers.

Figure 1.1: Top level flow chart for ode solver

This diagram illustrate some of the plots generated for direction field and phase plots.

Figure 1.2: Direction and slope fields generated