Consider a system model given by

fraction numerator d squared x left parenthesis t right parenthesis over denominator d t squared end fraction equals t
x left parenthesis 0 right parenthesis equals 1
right enclose fraction numerator d x left parenthesis t right parenthesis over denominator d t end fraction end enclose subscript t equals 0 end subscript equals 2.

What is the free response?

2 t plus 1

1 over 6 t cubed

1 over 6 t cubed plus 2 t plus 1

1

1 over 6 t cubed plus 2 t

2 t cubed plus 3 t plus 1

To solve for the free response of the system described by the second-order differential equation:

\[
\frac{d^2x(t)}{dt^2} = t
\]

with the initial conditions:

\[
x(0) = 1 \quad \text{and} \quad \frac{dx(t)}{dt} \Big|_{t=0} = 2,
\]

we proceed step by step.

### Step 1: Integrate the given differential equation.
First, integrate the equation \(\frac{d^2x(t)}{dt^2} = t\) with respect to \(t\):

\[
\frac{dx(t)}{dt} = \int t \, dt = \frac{t^2}{2} + C_1,
\]

where \(C_1\) is the constant of integration.

### Step 2: Use the initial velocity condition.
From the initial condition \(\frac{dx(t)}{dt} \Big|_{t=0} = 2\), substitute \(t = 0\) into the equation for \(\frac{dx(t)}{dt}\):

\[
2 = \frac{0^2}{2} + C_1 \quad \Rightarrow \quad C_1 = 2.
\]

Thus, the velocity expression becomes:

\[
\frac{dx(t)}{dt} = \frac{t^2}{2} + 2.
\]

### Step 3: Integrate again to find \(x(t)\).
Now integrate the expression for \(\frac{dx(t)}{dt}\) to find \(x(t)\):

\[
x(t) = \int \left(\frac{t^2}{2} + 2\right) dt = \frac{t^3}{6} + 2t + C_2,
\]

where \(C_2\) is another constant of integration.

### Step 4: Use the initial position condition.
From the initial condition \(x(0) = 1\), substitute \(t = 0\) into the expression for \(x(t)\):

\[
1 = \frac{0^3}{6} + 2(0) + C_2 \quad \Rightarrow \quad C_2 = 1.
\]

Thus, the solution for \(x(t)\) is:

\[
x(t) = \frac{t^3}{6} + 2t + 1.
\]

### Conclusion:
The free response of the system is:

\[
x(t) = \frac{t^3}{6} + 2t + 1.
\]

This matches the option:

\[
\boxed{\frac{t^3}{6} + 2t + 1}.
\]

Would you like further details or clarification?

Here are 5 related questions:
1. How do initial conditions affect the solution of differential equations?
2. What is the significance of the free response in a system model?
3. How do you solve a second-order differential equation with non-constant coefficients?
4. What methods can be used to solve higher-order differential equations?
5. How do you verify the correctness of a solution to a differential equation?

**Tip:** Always check initial conditions after solving to confirm the constants of integration are correct.

Consider a system model given by d²x(t)/dt² = t, with initial conditions x(0) = 1 and dx(t)/dt |t=0 = 2. What is the free response?

Solve a second-order differential equation d²x(t)/dt² = t with initial conditions x(0) = 1 and dx(t)/dt |t=0 = 2. Step-by-step solution involving integration and initial condition verification, ideal for undergraduate students studying differential equations.

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