To solve the differential equation:

$(D^2 + 3D + 2)y(t) = Dx(t)$

with the input $x(t) = 10e^{-3t}$ and initial conditions $y(0) = 2$ and $y'(0) = 3$, we can proceed as follows:

Step 1: Rewrite the Differential Operator

The operator form can be rewritten as: $\frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = \frac{dx}{dt}$

Step 2: Find the Homogeneous Solution

The characteristic equation for the homogeneous equation $\frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = 0$ is: $r^2 + 3r + 2 = 0$ Solving this, we find roots $r = -1$ and $r = -2$, so the homogeneous solution is: $y_h(t) = C_1 e^{-t} + C_2 e^{-2t}$

Step 3: Find the Particular Solution

To find a particular solution $y_p(t)$, we substitute the input $x(t) = 10e^{-3t}$ into the right-hand side: $\frac{dx}{dt} = -30e^{-3t}$ Thus, we are solving: $\frac{d^2y}{dt^2} + 3\frac{dy}{dt} + 2y = -30e^{-3t}$

Assuming a solution of the form $y_p(t) = Ae^{-3t}$ and substituting it into the equation, we can solve for $A$.

Step 4: Apply Initial Conditions

Combine the homogeneous and particular solutions, then use the initial conditions $y(0) = 2$ and $y'(0) = 3$ to solve for $C_1$ and $C_2$.

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Related Questions

1. How can you verify the particular solution for this differential equation?
2. What alternative methods exist for solving this type of differential equation?
3. How would the solution change if the input $x(t)$ were different?
4. What does the characteristic equation tell us about the system's stability?
5. How do initial conditions influence the form of the solution?

Tip

When solving differential equations, always start by solving the homogeneous equation before tackling the particular solution.