This week’s work involved extending the capabilities of dsolve_system by introducing a method to solve a particular form of second order system of ODEs.

Linear 2 equations, Order 2, Type 5, 8, 11

These systems are of the following format:

X'' = A(t)*(t*X' - X) + b(t)

Here, X is a vector of dependent variables, A(t) is a coefficient matrix and b(t) is the non-homogeneous term. To solve the above system for any number of equations, we will make a substitution and reduce the above system into a first order system of ODEs:

U = t*X' - X => U' = t*X''

Hence, the system becomes:

U' = t*A(t)*U + t*b(t)

Let’s say we solve for U. Now, we can easily solve for X by leveraging the substitution:

t*X' - X = U(t)

This is a solvable system of ODEs. This solution was implemented in this week and the old solvers that are being covered by this substitution are removed.

For the next week, a number of solvers are going to be implemented.