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LFCFAN
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Homework Statement
Why does the following ODE ALWAYS have two linearly independent solutions?
x''(t) + a(t) x'(t) + b(t) x(t) = f(t)
The characteristic polynomial argument is not sufficient?
This is because a 2nd-order linear ODE can be written in the form of a linear combination of two independent solutions. Any further solutions can be expressed as a linear combination of these two independent solutions, making them dependent on the first two solutions.
This limitation in the number of independent solutions allows us to easily determine a unique solution to the ODE by specifying only two initial or boundary conditions. It simplifies the solving process and makes it more manageable.
Yes, it is possible for a 2nd-order linear ODE to have only one independent solution. This occurs when the two roots of the characteristic equation are the same, resulting in a repeated solution. However, this is still considered to be at most two independent solutions.
Yes, there are some special cases where a 2nd-order linear ODE may have more than two independent solutions. This can happen when the coefficients of the ODE are variable functions or when it is a non-homogeneous ODE with a particular solution added to the general solution.
The number of independent solutions in a 2nd-order linear ODE can affect its stability in the sense that a larger number of independent solutions can result in a less stable system. This is because a higher number of solutions means there are more ways for the system to behave, making it more unpredictable and potentially less stable.