Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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**Section 4.2.11: Solving a Linear System of Differential Equations**

---

Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to \( t \).

\[
2x' + y' - 9x - 7y = e^{-t}
\]
\[
x' + y' + 4x + 3y = e^t
\]

**Step 1: Eliminate \( y \) and solve the remaining differential equation for \( x \). Choose the correct answer below:**

A. \( x(t) = C_1 \, e^t \cos(t) + C_2 \, e^t \sin(t) + 3 \, e^{-t} e^t \)

B. \( x(t) = C_1 \, e^t + C_2 \, e^{-t} + e^{-t} - 3 \, e^t \)

C. \( x(t) = C_1 \, \cos(t) + C_2 \, \sin(t) \)

D. \( x(t) = C_1 \, \cos(t) + C_2 \, \sin(t) + e^{-t} + 3 \, e^t \) *(Correct Answer)*

---

**Step 2: Determine \( y(t) **

Find \( y(t) \) such that \( y(t) \) and the solution for \( x(t) \) found in the previous step are a general solution to the system of differential equations.

\[ 
y(t) = \_\_\_
\] 

---

This solution involves solving the system of differential equations using elimination and selecting the correct general solution for \( x(t) \). Further calculations are needed to find \( y(t) \).
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Transcribed Image Text:**Section 4.2.11: Solving a Linear System of Differential Equations** --- Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to \( t \). \[ 2x' + y' - 9x - 7y = e^{-t} \] \[ x' + y' + 4x + 3y = e^t \] **Step 1: Eliminate \( y \) and solve the remaining differential equation for \( x \). Choose the correct answer below:** A. \( x(t) = C_1 \, e^t \cos(t) + C_2 \, e^t \sin(t) + 3 \, e^{-t} e^t \) B. \( x(t) = C_1 \, e^t + C_2 \, e^{-t} + e^{-t} - 3 \, e^t \) C. \( x(t) = C_1 \, \cos(t) + C_2 \, \sin(t) \) D. \( x(t) = C_1 \, \cos(t) + C_2 \, \sin(t) + e^{-t} + 3 \, e^t \) *(Correct Answer)* --- **Step 2: Determine \( y(t) ** Find \( y(t) \) such that \( y(t) \) and the solution for \( x(t) \) found in the previous step are a general solution to the system of differential equations. \[ y(t) = \_\_\_ \] --- This solution involves solving the system of differential equations using elimination and selecting the correct general solution for \( x(t) \). Further calculations are needed to find \( y(t) \).
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