Find a first order system of ordinary differential equations...

Find a first order system of ordinary differential equations that has the indicated vector-valued function as a solution: (a) $\left(\begin{array}{c}e^t+e^{2 t} \\ 2 e^t\end{array}\right)$, (b) $\left(\begin{array}{c}e^{-t} \cos 3 t \\ -3 e^{-t} \sin 3 t\end{array}\right)$, (c) $\left(\begin{array}{c}1 \\ t-1\end{array}\right)$, (d) $\left(\begin{array}{c}\sin 2 t-\cos 2 t \\ \sin 2 t+3 \cos 2 t\end{array}\right)$, (e) $\left(\begin{array}{c}e^{2 t} \\ e^{-3 t} \\ e^{2 t}-e^{-3 t}\end{array}\right)$, (f) $\left(\begin{array}{c}\sin t \\ \cos t \\ 1\end{array}\right)$, (g) $\left(\begin{array}{c}t \\ 1-t^2 \\ 1+t\end{array}\right)$, (h) $\left(\begin{array}{c}e^t \sin t \\ 2 e^t \cos t \\ e^t \sin t\end{array}\right)$.

Find a first order system of ordinary differential equations that has the indicated vector-valued function as a solution:
(a) $\left(\begin{array}{c}e^t+e^{2 t} \\ 2 e^t\end{array}\right)$,
(b) $\left(\begin{array}{c}e^{-t} \cos 3 t \\ -3 e^{-t} \sin 3 t\end{array}\right)$,
(c) $\left(\begin{array}{c}1 \\ t-1\end{array}\right)$,
(d) $\left(\begin{array}{c}\sin 2 t-\cos 2 t \\ \sin 2 t+3 \cos 2 t\end{array}\right)$,
(e) $\left(\begin{array}{c}e^{2 t} \\ e^{-3 t} \\ e^{2 t}-e^{-3 t}\end{array}\right)$,
(f) $\left(\begin{array}{c}\sin t \\ \cos t \\ 1\end{array}\right)$,
(g) $\left(\begin{array}{c}t \\ 1-t^2 \\ 1+t\end{array}\right)$,
(h) $\left(\begin{array}{c}e^t \sin t \\ 2 e^t \cos t \\ e^t \sin t\end{array}\right)$.

Key Concepts

First Order Differential Equations

A first order differential equation involves only the first derivative of the unknown function. In the context of systems, multiple first order equations are written together using a vector of dependent variables. Such systems describe the evolution of several interdependent quantities and are fundamental in modeling real-world processes where the rate of change depends on the current state.

Vector-Valued Functions

A vector-valued function assigns a vector to each value of a variable (typically time). In differential equations, these functions represent states in a multidimensional space where each component corresponds to a specific quantity. Analyzing vector-valued solutions is essential for understanding the dynamics of the entire system under study.

Matrix Representation of Systems

Systems of first order linear differential equations are often expressed in the compact form x' = Ax, where 'x' is a vector-valued function and 'A' is a constant or variable coefficient matrix. This representation allows for a systematic examination of the system's behaviour, including aspects like stability, eigenvalues, and phase plane analysis.

Constructing Differential Equations from Given Solutions

When a specific vector-valued function is known to be a solution, one can reverse-engineer a system by differentiating the solution components and expressing the derivatives as linear combinations of the original components. This process involves identifying relationships and possibly constructing an appropriate coefficient matrix so that the given function satisfies the differential system.

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