## Chapter 1Introduction

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.

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