Consider the following. d²y dt2 d²y d?x + dt2 d²x 6t dt2 x(0) = 7, x(0) = 0, y(0) = 0, y'(0) = 0 II
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Hello,
I asked for help on this question yesterday and was given the incorrect answers. I am attaching a copy of the problem. Your assistance is appreciated.
![**Consider the following:**
\[
\frac{d^2x}{dt^2} + \frac{d^2y}{dt^2} = t^2
\]
\[
\frac{d^2x}{dt^2} - \frac{d^2y}{dt^2} = 6t
\]
\[
x(0) = 7, \quad x'(0) = 0, \quad y(0) = 0, \quad y'(0) = 0
\]
---
**Give the following derivatives in terms of \( t \):**
\[
\frac{d^2x}{dt^2} = \frac{1}{2}t^2 - 3t \quad \text{\huge{✗}}
\]
\[
\frac{d^2y}{dt^2} = \frac{t^2 + 6t}{2} \quad \text{\huge{✗}}
\]
---
**Use the Laplace transform to solve the given system of differential equations:**
\[
x(t) = 7 + \frac{1}{24}t^4 - \frac{1}{2}t^3 \quad \text{\huge{✗}}
\]
\[
y(t) = \frac{1}{24}t^4 + \frac{1}{2}t^3 \quad \text{\huge{✗}}
\]
---
**Explanation:**
- The problem begins by considering a system of differential equations involving second derivatives of \( x \) and \( y \).
- Initial conditions are provided for both \( x(t) \) and \( y(t) \).
- The task is to express the second derivatives of \( x \) and \( y \) in terms of \( t \).
- There is a section indicating the use of the Laplace transform to find solutions for \( x(t) \) and \( y(t) \).
**Note:** Each step in these calculations has been marked with an "✗" possibly indicating a need for verification or completeness check.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c8a2351-0faf-49df-b50e-203220f7d5bc%2F5bb4657b-1fe4-4ec1-8db2-0909df43c080%2Fb5z005w_processed.png&w=3840&q=75)
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