We aim to solve the differential equation

$y'' - 2y' + y = \delta(t - 1), \quad y(0) = 3, \quad y'(0) = -4$

using the Laplace transform method.

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Steps:

1. Take the Laplace transform of the entire equation: Recall the Laplace transform rules:

   • $\mathcal{L}[y''(t)] = s^2 Y(s) - sy(0) - y'(0)$
   • $\mathcal{L}[y'(t)] = s Y(s) - y(0)$
   • $\mathcal{L}[y(t)] = Y(s)$
   • $\mathcal{L}[\delta(t - a)] = e^{-as}$

   Applying the Laplace transform to both sides of the equation: [ \mathcal{L}[y'' - 2y' + y] = \mathcal{L}[\delta(t - 1)] ]

   Using the initial conditions $y(0) = 3$ and $y'(0) = -4$: $(s^2 Y(s) - 3s + 4) - 2(s Y(s) - 3) + Y(s) = e^{-s}.$

2. Simplify the equation: Expand and group terms involving $Y(s)$: $s^2 Y(s) - 3s + 4 - 2s Y(s) + 6 + Y(s) = e^{-s}.$

   Combine like terms: $(s^2 - 2s + 1) Y(s) = e^{-s} + 3s - 10.$

   Simplify further: $(s - 1)^2 Y(s) = e^{-s} + 3s - 10.$

   Solve for $Y(s)$: $Y(s) = \frac{e^{-s} + 3s - 10}{(s - 1)^2}.$

3. Perform partial fraction decomposition: Decompose $\frac{3s - 10}{(s - 1)^2}$: $\frac{3s - 10}{(s - 1)^2} = \frac{A}{s - 1} + \frac{B}{(s - 1)^2}.$

   Solve for $A$ and $B$ by multiplying through by $(s - 1)^2$: $3s - 10 = A(s - 1) + B.$

   Expand and equate coefficients: $3s - 10 = As - A + B.$

   Comparing coefficients of $s$ and constant terms:

   • Coefficient of $s$: $A = 3$,
   • Constant term: $-A + B = -10$, so $-3 + B = -10$, giving $B = -7$.

   Thus: $\frac{3s - 10}{(s - 1)^2} = \frac{3}{s - 1} - \frac{7}{(s - 1)^2}.$

   Substitute back into $Y(s)$: $Y(s) = \frac{e^{-s}}{(s - 1)^2} + \frac{3}{s - 1} - \frac{7}{(s - 1)^2}.$

4. Take the inverse Laplace transform: Break $Y(s)$ into parts and find the inverse Laplace transform:

   • $\mathcal{L}^{-1}\left[\frac{3}{s - 1}\right] = 3e^t$,
   • $\mathcal{L}^{-1}\left[\frac{-7}{(s - 1)^2}\right] = -7te^t$,
   • $\mathcal{L}^{-1}\left[\frac{e^{-s}}{(s - 1)^2}\right] = u_1(t)(t - 1)e^{t - 1}$ (where $u_1(t)$ is the Heaviside function).

   Combine the terms: $y(t) = 3e^t - 7te^t + u_1(t)(t - 1)e^{t - 1}.$

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Final Solution:

$y(t) = 3e^t - 7te^t + u_1(t)(t - 1)e^{t - 1}.$

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Let me know if you need any steps clarified or further assistance! Here are related questions to deepen understanding:

1. What is the role of the Heaviside function in solutions involving impulses like $\delta(t - 1)$?
2. How does the Laplace transform simplify solving initial value problems?
3. Why is partial fraction decomposition important in this context?
4. What are the practical applications of impulse response in real-world systems?
5. How can we verify the solution using an alternative method like direct integration?

Tip: Always remember to correctly incorporate initial conditions when applying the Laplace transform, as they are critical in shaping the solution.