Question

First calculate the operational determinant of the given system in order to determine how many arbitrary constants should appear in a general solution. Then attempt to solve the system explicitly so as to find such a general solution. $\left(D^{2}+1\right) x+\left(D^{2}+2\right) y=2 e^{-t}$ $\left(D^{2}-1\right) x+D^{2} y=0$

          First calculate the operational determinant of the given system in order to determine how many arbitrary constants should appear in a general solution. Then attempt to solve the system explicitly so as to find such a general solution.
$\left(D^{2}+1\right) x+\left(D^{2}+2\right) y=2 e^{-t}$ $\left(D^{2}-1\right) x+D^{2} y=0$

Added by Cesar B.

Differential Equations and Linear Algebra

Stephen W. Goode, Scott A. Annin 4th Edition

Instant Answer

Solved by Expert Basques Licit

12/16/2021

Step 1

The operational determinant is given by $\left| \begin{array}{cc} D^2 + 1 & D^2 + 2 \\ D^2 - 1 & D^2 \end{array} \right| = (D^2 + 1)(D^2) - (D^2 + 2)(D^2 - 1) = D^4 - D^2 + D^2 - 2 = D^4 - 2$ Since the operational determinant is $D^4 - 2$, there are 2 arbitrary Show more…

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First calculate the operational determinant of the given system in order to determine how many arbitrary constants should appear in a general solution. Then attempt to solve the system explicitly so as to find such a general solution. $\left(D^{2}+1\right) x+\left(D^{2}+2\right) y=2 e^{-t}$ $\left(D^{2}-1\right) x+D^{2} y=0$