Use the Laplace transform to solve the given system of differential equations. d?x + x - y = 0 dt? d²y + y - x = 0 dt? x(0) = 0, x'(0) = -3, y(0) = 0, y'(0) = 1 %3D %3D
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![**Educational Content: Solving Differential Equations Using the Laplace Transform**
This example demonstrates how to use the Laplace transform to solve a system of differential equations.
**Given System of Differential Equations:**
1. \(\frac{d^2x}{dt^2} + x - y = 0\)
2. \(\frac{d^2y}{dt^2} + y - x = 0\)
**Initial Conditions:**
- \(x(0) = 0\)
- \(x'(0) = -3\)
- \(y(0) = 0\)
- \(y'(0) = 1\)
**Proposed Solutions:**
- The solution for \(x(t)\) is given as:
\[
x(t) = -\frac{1}{3}t - \frac{4}{3\sqrt{3}} \sin(\sqrt{3}t)
\]
(Indicated as incorrect by a red cross)
- The solution for \(y(t)\) is given as:
\[
y(t) = -\frac{1}{3}t + \frac{4}{3\sqrt{3}} \sin(\sqrt{3}t)
\]
(Indicated as incorrect by a red cross)
Each red cross indicates that the proposed solution did not satisfy the system of differential equations or initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c8a2351-0faf-49df-b50e-203220f7d5bc%2Fc60283dd-6349-4b4a-b368-69590618bc64%2Fhbnyowt_processed.png&w=3840&q=75)
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