Fundamental matrices in inhomogenous problems?

[pivoxa15]

It seems to be the case that fundamental matrices are only used when the particular solution has an initial value of 0. Since the integral for the inhomogenous term is from (t. to t) where t. is initial value. So when t=t. the inhomogenous term must be 0. But there are cases when the particlular solution may not have an intitial value of 0 which means those situations cannot be dealt with by Fundamental matrices method of solving ODEs. But I always had the impression that they could solve any systems of ODEs.

 

[gtoguy97]

Fundamental matrices can indeed solve any system of ODEs, regardless of the particular solution having an initial value of 0 or not. The fundamental matrix provides a general solution to a system of linear ODEs, and the particular solution can be found by solving for the constants of integration. The only difference between the case where the particular solution has an initial value of 0 and when it does not is that in the former case, the constants of integration can be easily determined from the initial condition, while in the latter case, the constants must be found by solving the system of equations given by the initial conditions.

 

[tacman]

Fundamental matrices can certainly be used to solve inhomogeneous problems, even when the particular solution does not have an initial value of 0. The key is to use the fundamental matrix to find the general solution, and then use the initial conditions to find the particular solution. In other words, the fundamental matrix provides the building blocks for finding the solution, but the initial conditions are still necessary to fully determine the particular solution.

In some cases, the inhomogeneous term may be 0 at the initial value, but this is not always the case. For example, consider the ODE y'' + y = sin(t), with initial conditions y(0) = a and y'(0) = b. The inhomogeneous term is not 0 at t=0, but we can still use the fundamental matrix to find the general solution y(t) = c1*cos(t) + c2*sin(t) + sin(t)/2. Then, using the initial conditions, we can solve for c1 and c2 and obtain the particular solution y(t) = a*cos(t) + (b+1)*sin(t)/2.

So while it may be true that the inhomogeneous term must be 0 at the initial value for the particular solution to have an initial value of 0, this does not mean that fundamental matrices cannot be used to solve inhomogeneous problems. They are still a powerful tool for solving systems of ODEs, and can handle a variety of initial conditions and inhomogeneous terms.