Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t. 2x' + y'-7x-6y = e-t x'+y' + 3x +2y = et Eliminate y and solve the remaining differential equation for x. Choose the correct answer below. 1 -t A. x(t) = C₁ cos (2t) + C₂ sin (2t) + 5 e + O B. x(t) = C₁ cos (2t) + C₂ sin (2t) 1 - 2t OC. x(t) = C₁ e ²t + + C₂ e 5 1 D. x(t) = C₁ e ²t cos (2t) + C₂ e ²¹ sin (2t) + e Now find y(t) so 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) = + e
Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t. 2x' + y'-7x-6y = e-t x'+y' + 3x +2y = et Eliminate y and solve the remaining differential equation for x. Choose the correct answer below. 1 -t A. x(t) = C₁ cos (2t) + C₂ sin (2t) + 5 e + O B. x(t) = C₁ cos (2t) + C₂ sin (2t) 1 - 2t OC. x(t) = C₁ e ²t + + C₂ e 5 1 D. x(t) = C₁ e ²t cos (2t) + C₂ e ²¹ sin (2t) + e Now find y(t) so 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) = + e
Calculus: Early Transcendentals
8th Edition
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Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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![### Solving Linear Systems of Differential Equations Using the Elimination Method
To derive a general solution for the given linear system, we begin by addressing the following equations, where differentiation is with respect to \( t \):
\[
\begin{cases}
2x' + y' - 7x - 6y = e^{-t} \\
x' + y' + 3x + 2y = e^t
\end{cases}
\]
Next, we eliminate \( y \) to solve the remaining differential equation for \( x \). You will be choosing the correct solution from the options below:
**Eliminate \( y \) and solve the remaining differential equation for \( x \). Choose the correct answer:**
- **A.** \( x(t) = C_1 \cos (2t) + C_2 \sin (2t) + \frac{1}{5}e^{-t} + e^t \) ✅
- **B.** \( x(t) = C_1 \cos (2t) + C_2 \sin (2t) \)
- **C.** \( x(t) = C_1 e^{2t} + C_2 e^{-2t} + \frac{1}{5}e^{-t} - e^{-t} + e^t \)
- **D.** \( x(t) = C_1 e^{2t} \cos (2t) + C_2 e^{2t} \sin (2t) + e^{-t} - \frac{1}{5} e^t \)
Chosen correctly, **Option A** gives the general solution for \( x(t) \).
Now, using your expression for \( x(t) \) and following the suitable steps, find \( y(t) \) so that \((x(t), y(t))\) encapsulates the general solution to the system of differential equations.
### Solve for \( y(t) \):
\[ y(t) = \]
---
This segment breaks down the process into logical steps and portrays a clear pathway for students to follow, facilitating their engagement with the material on an educational website. The equation solution is highlighted distinctly to ensure easy identification of the correct answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F896deee6-4ebc-4afd-8502-502eb7aa6712%2F583ba1a6-ca3f-449e-a87f-14a042bb94f1%2F9mnv04n_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Linear Systems of Differential Equations Using the Elimination Method
To derive a general solution for the given linear system, we begin by addressing the following equations, where differentiation is with respect to \( t \):
\[
\begin{cases}
2x' + y' - 7x - 6y = e^{-t} \\
x' + y' + 3x + 2y = e^t
\end{cases}
\]
Next, we eliminate \( y \) to solve the remaining differential equation for \( x \). You will be choosing the correct solution from the options below:
**Eliminate \( y \) and solve the remaining differential equation for \( x \). Choose the correct answer:**
- **A.** \( x(t) = C_1 \cos (2t) + C_2 \sin (2t) + \frac{1}{5}e^{-t} + e^t \) ✅
- **B.** \( x(t) = C_1 \cos (2t) + C_2 \sin (2t) \)
- **C.** \( x(t) = C_1 e^{2t} + C_2 e^{-2t} + \frac{1}{5}e^{-t} - e^{-t} + e^t \)
- **D.** \( x(t) = C_1 e^{2t} \cos (2t) + C_2 e^{2t} \sin (2t) + e^{-t} - \frac{1}{5} e^t \)
Chosen correctly, **Option A** gives the general solution for \( x(t) \).
Now, using your expression for \( x(t) \) and following the suitable steps, find \( y(t) \) so that \((x(t), y(t))\) encapsulates the general solution to the system of differential equations.
### Solve for \( y(t) \):
\[ y(t) = \]
---
This segment breaks down the process into logical steps and portrays a clear pathway for students to follow, facilitating their engagement with the material on an educational website. The equation solution is highlighted distinctly to ensure easy identification of the correct answer.
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