Find the general solution to the following forced second...

Find the general solution to the following forced second order systems: (a) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}7 & -2 \\ -2 & 4\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 0\end{array}\right)$, (b) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}5 & -2 \\ -2 & 3\end{array}\right) \mathbf{u}=\left(\begin{array}{c}0 \\ 5 \sin 3 t\end{array}\right)$, (c) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}13 & -6 \\ -6 & 8\end{array}\right) \mathbf{u}=\left(\begin{array}{r}5 \cos 2 t \\ \cos 2 t\end{array}\right)$, (d) $\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right) \frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}3 & -1 \\ -1 & 2\end{array}\right) \mathbf{u}=\left(\begin{array}{r}\cos \frac{1}{2} t \\ -\cos \frac{1}{2} t\end{array}\right)$. (e) $\left(\begin{array}{ll}3 & 0 \\ 0 & 5\end{array}\right) \frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}4 & -2 \\ -2 & 3\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 11 \sin 2 t\end{array}\right)$, (f) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 0 \\ \cos t\end{array}\right)$

Find the general solution to the following forced second order systems:
(a) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}7 & -2 \\ -2 & 4\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 0\end{array}\right)$,
(b) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}5 & -2 \\ -2 & 3\end{array}\right) \mathbf{u}=\left(\begin{array}{c}0 \\ 5 \sin 3 t\end{array}\right)$,
(c) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}13 & -6 \\ -6 & 8\end{array}\right) \mathbf{u}=\left(\begin{array}{r}5 \cos 2 t \\ \cos 2 t\end{array}\right)$,
(d) $\left(\begin{array}{ll}2 & 0 \\ 0 & 3\end{array}\right) \frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}3 & -1 \\ -1 & 2\end{array}\right) \mathbf{u}=\left(\begin{array}{r}\cos \frac{1}{2} t \\ -\cos \frac{1}{2} t\end{array}\right)$.
(e) $\left(\begin{array}{ll}3 & 0 \\ 0 & 5\end{array}\right) \frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rr}4 & -2 \\ -2 & 3\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 11 \sin 2 t\end{array}\right)$,
(f) $\frac{d^2 \mathbf{u}}{d t^2}+\left(\begin{array}{rrr}6 & -4 & 1 \\ -4 & 6 & -1 \\ 1 & -1 & 11\end{array}\right) \mathbf{u}=\left(\begin{array}{c}\cos t \\ 0 \\ \cos t\end{array}\right)$

Key Concepts

Eigenvalue and Eigenvector Analysis

In the context of coupled systems, eigenvalue analysis is used on the coefficient (stiffness) matrix to determine the natural frequencies and modes of the system. This analysis allows for decoupling the system by transforming it into a coordinate system formed by the eigenvectors, turning a complex coupled problem into simpler independent equations.

Method of Undetermined Coefficients

This technique is used to find a particular solution of non-homogeneous differential equations when the forcing functions are of a specific form (like sines, cosines, or polynomials). It involves assuming a form a priori for the particular solution with unknown coefficients that are determined by substituting back into the differential equation.

System Decoupling and Diagonalization

Decoupling is achieved by diagonalizing the system matrices through an appropriate transformation, often via the system’s eigenvectors. This process simplifies solving the system by reducing it to several independent second-order differential equations, one for each decoupled mode.

Second-Order Linear Differential Systems

This concept involves differential equations where the highest derivative is of the second order and the system may consist of multiple coupled equations. Understanding these systems is central to modeling many physical phenomena, and they are expressed compactly using matrix notation to encapsulate the interdependencies between multiple variables.

Homogeneous and Particular Solutions

The overall solution to a forced differential system is constructed by combining the general solution of the homogeneous part (where the forcing term is zero) with a particular solution that accounts for the external forcing. The homogeneous solution typically involves finding the characteristic equation and using its roots, while the particular solution is obtained by techniques such as the method of undetermined coefficients.

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