Find the general real solution to the following systems of...

Find the general real solution to the following systems of differential equations: (a) $$ \begin{aligned} & \dot{u}_1=u_1+9 u_2 \\ & \dot{u}_2=u_1+3 u_2 \\ & \dot{y}_1=y_2 \\ & \dot{y}_2=3 y_1+2 y_3 \\ & \dot{y}_3=-y_2 \end{aligned} $$ (d) $$ \begin{aligned} & \dot{y}_2=3 y_1+2 y_3 \\ & \dot{y}_3=-y_2 \end{aligned} $$ (b) $$ \begin{aligned} & \dot{x}_1=4 x_1+3 x_2, \quad \text { (c) } \begin{array}{l} \dot{y}_1=y_1-y_2, \\ \dot{y}_2=2 y_1+3 y_2 ; \\ \dot{x}_2=3 x_1-4 x_2 ; \end{array} \dot{x}_1=3 x_1-8 x_2+2 x_3, \quad \dot{u}_1=u_1-3 u_2+11 u_3, \end{aligned} $$ (c) $$ \dot{y}_1=y_1-y_2 $$ (e) $$ \begin{aligned} & \dot{x}_2=-x_1+2 x_2+2 x_3, \\ & \dot{x}_3=x_1-4 x_2+2 x_3 \end{aligned} $$ (f) $$ \begin{aligned} & \dot{u}_2=2 u_1-6 u_2+16 u_3, \\ & \dot{u}_3=u_1-3 u_2+7 u_3 . \end{aligned} $$

Find the general real solution to the following systems of differential equations:
(a)
$$
\begin{aligned}
& \dot{u}_1=u_1+9 u_2 \\
& \dot{u}_2=u_1+3 u_2 \\
& \dot{y}_1=y_2 \\
& \dot{y}_2=3 y_1+2 y_3 \\
& \dot{y}_3=-y_2
\end{aligned}
$$
(d)
$$
\begin{aligned}
& \dot{y}_2=3 y_1+2 y_3 \\
& \dot{y}_3=-y_2
\end{aligned}
$$
(b)
$$
\begin{aligned}
& \dot{x}_1=4 x_1+3 x_2, \quad \text { (c) } \begin{array}{l}
\dot{y}_1=y_1-y_2, \\
\dot{y}_2=2 y_1+3 y_2 ; \\
\dot{x}_2=3 x_1-4 x_2 ;
\end{array} \dot{x}_1=3 x_1-8 x_2+2 x_3, \quad \dot{u}_1=u_1-3 u_2+11 u_3,
\end{aligned}
$$
(c)
$$
\dot{y}_1=y_1-y_2
$$
(e)
$$
\begin{aligned}
& \dot{x}_2=-x_1+2 x_2+2 x_3, \\
& \dot{x}_3=x_1-4 x_2+2 x_3
\end{aligned}
$$
(f)
$$
\begin{aligned}
& \dot{u}_2=2 u_1-6 u_2+16 u_3, \\
& \dot{u}_3=u_1-3 u_2+7 u_3 .
\end{aligned}
$$

Key Concepts

Systems of Linear Differential Equations

This concept involves a set of equations where the derivatives of unknown functions are expressed as linear combinations of those functions. The overall goal is to find the general solution, which typically involves methods like eigen-decomposition or the use of the matrix exponential to capture the system's dynamic behavior.

Eigenvalue and Eigenvector Analysis

A key method for solving linear systems is to find the eigenvalues and corresponding eigenvectors of the coefficient matrix. These eigenpairs allow one to express the general solution as a linear combination of exponential terms, where each term is scaled by an eigenvalue and directed by its eigenvector.

Decoupling and Block Diagonal Structure

Often, systems of differential equations can be partitioned into smaller subsystems that are either fully or partially decoupled. Recognizing such structures simplifies the solution process since each subsystem can be handled independently, reducing a complex problem into a set of simpler ones.

Real Solutions from Complex Eigenvalues

When the eigenvalue analysis yields complex eigenvalues, the general solution is initially expressed in complex form. However, for real systems, these complex solutions must be transformed into a real formulation using techniques that exploit the symmetry of complex conjugate pairs, often resulting in sine and cosine functions.

Superposition Principle

For linear systems, any linear combination of solutions is itself a solution. This principle underpins the construction of the general solution, as it allows the combination of independent solution components, each associated with an eigenvalue, to form the complete response to the system.

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