I am given a 6th order homogenous differential equation and I am tasked to find the dimensions of the linear space of solutions to the differential equation.
What would be the theory that I need to know to solve this question? Is the dimensions just the number of roots to the equation minus one repeats?
For a linear differential equation you have as many dimensions as you have independent solutions to the (homogeneous) equation. The is often the degree of the equation, so six here. It works if the equation is linear in the dependent variable and its derivatives. The idea is that you can multiply any solution by a constant to get a new solution or add any two solutions to get a new solution. That is the defining property of a vector space. As you can multiply by any constant, you can multiply by zero to get a solution that is constantly zero. That is the identity element of the space.