The first step is to obtain the general solution, which will contain arbitrary number of unknown constants. The number of IC’s given can be one, two or more depending on the order of the ode. Any is allowed.
For each IC, we set up one equation by plugging the general solution into the IC. We do this for each IC given. For example, if two IC’s are given, we end up with two equations.
Now we solve these equations for the unknown constants of integrations.
There are three possible cases. There can be more equations than the constants (this means more IC’s than the order of the ode). The same number of equations as the constants, or smaller number of equations than the constants.
For the case where there are more equations than the constants, this is called overdetermined system. For example, if the ode is first order, and we have given two different IC’s. In most cases there will be no solution for this case. We have to use linear algebra to determine this. Only if the two equations are scaled version of each other will there be solution.
For the case when the same number of equations as the constants, this is the typical case and can be solved assuming IC’s are valid.
For the case when there is less number of equations than the constants, where this happens for example when the ode is say a second order and there is only one IC, then in this case one or more of the constants will remain unresolved and the other constants are solved for in terms of the others. These cases are now illustrated using examples below.
There are cases when the above main algorithm fails. In this case secondary algorithm is used. See section below.