We are given a system model:

$2 \frac{d^2 x(t)}{dt^2} + 3 \frac{dx(t)}{dt} + x(t) = \delta(t)$ with the initial conditions: $x(0^-) = 0, \quad \left. \frac{dx(t)}{dt} \right|_{t=0^-} = 1.$

Step 1: Analyze the Differential Equation

The system is a second-order linear differential equation with an impulse input $\delta(t)$. The impulse input suggests that there will be a discontinuity or jump in the derivative of $x(t)$ at $t=0$.

Step 2: Apply Laplace Transform

To handle the discontinuity, we apply the Laplace transform to both sides of the equation. Using the fact that:

$\mathcal{L}\left\{ \frac{d^2x(t)}{dt^2} \right\} = s^2 X(s) - s x(0^-) - \left. \frac{dx(t)}{dt} \right|_{t=0^-},$ $\mathcal{L}\left\{ \frac{dx(t)}{dt} \right\} = s X(s) - x(0^-),$ $\mathcal{L}\left\{ x(t) \right\} = X(s),$ and $\mathcal{L}\{\delta(t)\} = 1$, we transform the differential equation:

$2(s^2 X(s) - s x(0^-) - \left. \frac{dx(t)}{dt} \right|_{t=0^-}) + 3(s X(s) - x(0^-)) + X(s) = 1.$

Step 3: Substitute Initial Conditions

We are given $x(0^-) = 0$ and $\left. \frac{dx(t)}{dt} \right|_{t=0^-} = 1$. Substituting these into the Laplace-transformed equation:

$2(s^2 X(s) - 1) + 3(s X(s)) + X(s) = 1.$

Simplifying:

$2s^2 X(s) - 2 + 3s X(s) + X(s) = 1,$ $(2s^2 + 3s + 1) X(s) = 3,$ $X(s) = \frac{3}{2s^2 + 3s + 1}.$

Step 4: Inverse Laplace Transform

Now, we take the inverse Laplace transform to find $x(t)$. The Laplace transform of the system suggests a step response for a second-order system with an impulse input. To determine the initial conditions at $t = 0^+$, we use the fact that the system responds to the impulse input.

Since $x(t)$ was initially at rest ($x(0^-) = 0$), and there was a discontinuity in $\frac{dx(t)}{dt}$, we know:

$x(0^+) = x(0^-) = 0.$

To find $\left. \frac{dx(t)}{dt} \right|_{t=0^+}$, we analyze the jump condition caused by the impulse input. The impulse $\delta(t)$ forces an immediate change in the derivative. Given that $\left. \frac{dx(t)}{dt} \right|_{t=0^-} = 1$, and considering the dynamics of the system, we expect the initial derivative to remain the same:

$\left. \frac{dx(t)}{dt} \right|_{t=0^+} = 1.$

Final Answer:

The correct initial conditions are:

$x(0^+) = 0, \quad \left. \frac{dx(t)}{dt} \right|_{t=0^+} = 1.$

Thus, the correct choice is: $\boxed{x(0^+) = 0 \text{ and } \frac{dx}{dt}(0^+) = 1.}$

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Would you like further details on the steps, or do you have any questions? Here are some follow-up questions that might interest you:

1. How does the Laplace transform handle discontinuities in time-domain functions?
2. Why does an impulse input result in a jump in the system’s derivative?
3. What role do initial conditions play in solving differential equations using Laplace transforms?
4. How can we interpret second-order system behavior from its differential equation?
5. What methods can be used to find inverse Laplace transforms of rational functions?

Tip: Always remember that the Laplace transform is a powerful tool for solving differential equations, especially when dealing with initial conditions and discontinuous inputs like impulses.