## Introduction

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

Each ode type is given an internal code name. This internal code is used internally by the solver to determine which solver to call to solve the speciﬁc 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 linear in $$y'(x)$$.
2. First order ode not linear in $$y'(x)$$ (such as d’Alembert, 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.

For a diﬀerential equation, there are three types of solutions

1. General solution. This is the solution $$y(x)$$ which contains arbitrary number of constants up to the order of the ode.
2. Particular solution. This is the general solution after determining speciﬁc values for the constant of integrations from the given initial or boundary conditions. This solution will then contain no arbitrary constants.
3. singular solutions. These are solutions to the ode which satisfy the ode itself and contain no arbitrary constants but can not be found from the general solution using any speciﬁc values for the constants of integration. These solutions are found using diﬀerent methods than those used to ﬁnding the general solution. Singular solution are hence not found from the general solution like the case is with particular solution.