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.
First order ode.
Second and higher order ode.
For first order ode, the following are the main classifications used.
First order ode linear in \(y'(x)\).
First order ode not linear in \(y'(x)\) (such as d’Alembert, Clairaut).
For second order and higher order ode’s, further classification is
Linear ode.
non-linear ode.
Another classification for second order and higher order ode’s is
Constant coefficients ode.
Varying coefficients ode
Another classification for second order and higher order ode’s is
Homogeneous ode. (the right side is zero).
Non-homogeneous ode. (the right side is not zero).
All of the above can be combined to give this classification
First order ode.
First order ode linear in \(y'(x)\).
First order ode not linear in \(y'(x)\) (such as d’Alembert, Clairaut).
Second and higher order ode
Linear second order ode.
Linear homogeneous ode. (the right side is zero).
Linear homogeneous and constant coefficients ode.
Linear homogeneous and non-constant coefficients ode.
Linear non-homogeneous ode. (the right side is not zero).
Linear non-homogeneous and constant coefficients ode.
Linear non-homogeneous and non-constant coefficients ode.
Nonlinear second order ode.
Nonlinear homogeneous ode.
Nonlinear non-homogeneous ode.
For system of differential equation the following classification is used.
System of first order odes.
Linear system of odes.
non-linear system of odes.
System of second order odes.
Linear system of odes.
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
General solution. This is the solution \(y(x)\) which contains arbitrary number of
constants up to the order of the ode.
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.
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.