Homogeneous differential equations of arbitrary order with constant coefficients can be solved in straightforward matter by converting them into system of first order ODEs.
Example:
Example. Consider ODE of order 3:
Consider a second order vector differential equation
\[
\ddot{\bf x} + {\bf A} \, {\bf x} = {\bf 0} ,
\]
where
A is a positive
\( n \times n \) square matrix (which means that all its eigenvalues are positive) and
x(t) is a colomn-vector of unknown functions to be determined.
The general solution
\[
{\bf x}(t) = {\bf \Psi}(t)\,{\bf d} + {\bf \Phi}(t)\, {\bf v},
\]
is expressed through two fundamental matrices
\[
{\bf \Psi}(t) = \cos \left( \sqrt{\bf A}\, t\right) \qquad\mbox{and} \qquad {\bf \Phi}(t) = \dfrac{\sin \left( \sqrt{\bf A} \,t \right)}{\sqrt{\bf A}},
\]
and
d and
v are initial displacements and velocities, that is,
\( {\bf x}(0) = {\bf d} \) and
\( \dot{\bf x}(0) = {\bf v} .\) Since these two fundamental matrices are solutions of the same matrix differential equation, but distinct initial conditions:
\[
\ddot{\bf X} + {\bf A} \, {\bf X} = {\bf 0} , \qquad {\bf \Psi}(0) = {\bf I}, \quad \dot{\bf \Psi}(0) = {\bf 0}, \qquad {\bf \Phi}(0) = {\bf 0}, \quad \dot{\bf \Phi}(0) = {\bf I},
\]
The solution
\( {\bf x}(t) = {\bf \Psi}(t)\,{\bf d} + {\bf \Phi}(t)\, {\bf v} \) of the vector second order differential equation
\( \ddot{\bf x} (t) + {\bf A} \, {\bf x} = {\bf 0} \) satisfies the initial conditions:
\[
{\bf x}(0) = {\bf d} \qquad\mbox{and} \qquad \dot{\bf x}(0) = {\bf v} .
\]
This problem can also be solved by converting the given second order vector differential equation to the system of first order differential equations
\[
\dot{\bf y} = {\bf B}\,{\bf y} ,
\]
where
\( {\bf y}(t) = \langle {\bf x}, \dot{\bf x} \rangle \) is 2n column vector of displace ments and velocities, and B is a corresponding
\( 2n \times 2n \) matrix. However, this approach is not optimal and requires more efforts, as it will be shown by examples.
Example 2.2.1:
Consider the second order vector differential equation
\( \ddot{\bf x} + {\bf A}\,{\bf x} = {\bf 0} ,\) where
\[
\left[ \begin{array}{cc}
16& -9 \\ -12 & 13 \end{array} \right] .
\]
First, we find its eigenvalues and eigenvectors:
A:=matrix(2,2,[16,-9,-12,13])
\(
\left(\begin{array}{cc} 16 & -9\\ -12 & 13 \end{array}\right)
\)
linalg:: eigenvalues(A)
\(
\left\{4,25\right\} \)
So the matrix
A has two positive eigenvalues 4 and 25 (such
matrices are called positive).
To solve the given system of differentiql equations of the second
order, we need to construct two fundamental matrices
\[
{\bf \Psi}(t) = \cos \left( \sqrt{\bf A}\, t\right) \qquad\mbox{and} \qquad {\bf \Phi}(t) = \dfrac{\sin \left( \sqrt{\bf A} \,t \right)}{\sqrt{\bf A}},
\]
MuPad is so powerful that you can get these matrices with simple
commands:
Psi(t):=cos(sqrt(A)*t)
\(
\left(\begin{array}{cc}
\frac{3\,\cos\left(2\,t\right)}{7}+\frac{4\,\cos\left(5\,t\right)}{7}
&
\frac{3\,\cos\left(2\,t\right)}{7}-\frac{3\,\cos\left(5\,t\right)}{7}\\
\frac{4\,\cos\left(2\,t\right)}{7}-\frac{4\,\cos\left(5\,t\right)}{7}
&
\frac{4\,\cos\left(2\,t\right)}{7}+\frac{3\,\cos\left(5\,t\right)}{7}
\end{array}\right) \)
Phi(t):= sin(sqrt(A)*t)/sqrt(A)
\(
\left(\begin{array}{cc} \frac{3\,\sin\left(2\,t\right)}{14}+\frac{4\,\sin\left(5\,t\right)}{35} & \frac{3\,\sin\left(2\,t\right)}{14}-\frac{3\,\sin\left(5\,t\right)}{35}\\ \frac{2\,\sin\left(2\,t\right)}{7}-\frac{4\,\sin\left(5\,t\right)}{35} & \frac{2\,\sin\left(2\,t\right)}{7}+\frac{3\,\sin\left(5\,t\right)}{35} \end{array}\right) \)
Example 2.2.2:
Consider two mass system connected with three spings governed by Hooke's law. Then such mechanical system can be modeled by the following system of ordinary differential equations:
\[
\begin{cases}
m_1 \, \dfrac{{\text d}^2 x_1}{{\text d}\,t^2} + \left( k_1 + k_2 \right) x_1 - k_2 \,x_2 &=0 , \\
m_2 \, \dfrac{{\text d}^2 x_2}{{\text d}\,t^2} + \left( k_2 + k_3 \right) x_2 - k_3 \,x_1 &=0 , \\
\end{cases} \qquad \begin{bmatrix} x_1 (0) \\ x_2 (0) \end{bmatrix} = \begin{bmatrix} d_1 \\ d_2 \end{bmatrix} , \qquad
\begin{bmatrix} \dot{x}_1 (0) \\ \dot{x}_2 (0) \end{bmatrix} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} ,
\]
where
\( x_1 (t) \) and
\( x_2 (t) \) are displacements of masses
\( m_1 \) and
\( m_1 ,\) respectively. Here
\( k_1 , \ k_2 , \ k_3 \) are spring constants, and
\( d_1 , \ d_2 \) are initial displacements,
\( v_1 , \ v_2 \) are initial velocities.