This answer was marked wrong. It requires both x(t) and y(t) and any combination of what you wrote is being marked wrong by cengage. ### Solving Systems of Differential Equations by Systematic EliminationThe goal is to solve the given system of differential equations by systematic elimination. The system is:\[\frac{d^2 x}{dt^2} = 16 y + e^t\]\[\frac{d^2 y}{dt^2} = 16 x - e^t\]To find the solutions \( x(t) \) and \( y(t) \), we can proceed as follows:1. Define the general solution structure for both \( x(t) \) and \( y(t) \) by combining terms that arise from solving the homogeneous parts of each differential equation and particular solutions that address the non-homogeneous parts (i.e., terms involving \( e^t \)).2. Apply systematic elimination techniques to decouple the equations where possible.The solutions are then given in the following closed form:\[(x(t), y(t)) = \left( C_1 e^{-4t} + C_2 e^{4t} + C_1 \cos(4t) + C_2 \sin(4t), \, C_1 e^{-4t} + C_2 e^{4t} - C_1 \cos(4t) - C_2 \sin(4t) + \frac{e^t}{17} \right)\]Here:- \( C_1 \) and \( C_2 \) are constants determined by initial conditions.- The solution incorporates exponential and trigonometric functions indicating characteristic roots.- The term \(\frac{e^t}{17}\) is a particular solution addressing the non-homogeneous part of the system.**Need Help?** Click [Read It] for an interactive guide and detailed step-by-step breakdown.![Need Help Button](ReadItButton.png)

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Answered: Solve the given system of differential…

Math

Advanced Math

Solve the given system of differential equations by systematic elimination. d²x dt² d²y = 16x - et dt² = 16y + et

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### Solving Systems of Differential Equations by Systematic Elimination

The goal is to solve the given system of differential equations by systematic elimination. The system is:

\[
\frac{d^2 x}{dt^2} = 16 y + e^t
\]

\[
\frac{d^2 y}{dt^2} = 16 x - e^t
\]

To find the solutions \( x(t) \) and \( y(t) \), we can proceed as follows:

1. Define the general solution structure for both \( x(t) \) and \( y(t) \) by combining terms that arise from solving the homogeneous parts of each differential equation and particular solutions that address the non-homogeneous parts (i.e., terms involving \( e^t \)).

2. Apply systematic elimination techniques to decouple the equations where possible.

The solutions are then given in the following closed form:

\[
(x(t), y(t)) = \left( C_1 e^{-4t} + C_2 e^{4t} + C_1 \cos(4t) + C_2 \sin(4t), \, C_1 e^{-4t} + C_2 e^{4t} - C_1 \cos(4t) - C_2 \sin(4t) + \frac{e^t}{17} \right)
\]

Here:
- \( C_1 \) and \( C_2 \) are constants determined by initial conditions.
- The solution incorporates exponential and trigonometric functions indicating characteristic roots.
- The term \(\frac{e^t}{17}\) is a particular solution addressing the non-homogeneous part of the system.

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