Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations

Introduction

Translation:

We call an expression in the form

X' = A ⋅ X + G(t)

homogeneous if G(t)≡0. Now, in previous methods of differential equations, it turned out that X had an exponential of the transcendental number e in its form, so if a uniqueness theorem is developed, we can define a possible answer with this form, set it in the equation, and determine if this answer works and if so how to obtain the answer and its corresponding exponentials.

Results

So, because the exponential function appeared many times in simpler differential equations, we will guess that the solution for X is X = u ⋅ $e^{\lambda t}$, where u is a coefficient matrix.

Thus:

$\lambda \mathbf {u} e^{\lambda t}=\mathbf {A} \mathbf {u} e^{\lambda t}$

$\lambda e^{\lambda t}=\mathbf {A} e^{\lambda t}$

$(\mathbf {A} -\lambda \mathbf {I} )\mathbf {u} =0$

There is a lie here, we're also going to make one more assumption: a constant matrix for A; but this is the definition for an eigenvalue-eigenvector pair! Thus with a two-by-two matrix there are two linearly independent solutions, and thus by the principle of superposition the constant matrix multiplication by an augmented matrix of these two solutions makes the fundamental set of solutions of which we are trying to look for.

However, due to the property of these eigenvalues (and that we want real-solutions to help analysis in physical models utilizing these differential techniques), there are different ways of creating the fundamental set of solutions to the three possible cases that the pair of eigenvalues could fall under:

• Real, Distinct Eigenvalues Method

• Imaginary Eigenvalues Method

• Repeated Eigenvalue Method