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 forced response?

		
1 over 6 t cubed plus 2 t plus 1

		
2 t plus 1

		
1

		
1 over 6 t cubed plus 2 t

		
1 over 6 t cubed

		
2 t cubed plus 3 t plus 1



We are given the second-order differential equation:

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

with the initial conditions:

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

### Step 1: Solve the differential equation
To solve this, we need to integrate the equation twice to find \(x(t)\).

#### First integration
\[
\frac{d^2x(t)}{dt^2} = t
\]
Integrating both sides with respect to \(t\):
\[
\frac{dx(t)}{dt} = \int t \, dt = \frac{t^2}{2} + C_1,
\]
where \(C_1\) is a constant of integration.

#### Second integration
Now, integrate again to find \(x(t)\):
\[
x(t) = \int \left(\frac{t^2}{2} + C_1\right) dt = \frac{t^3}{6} + C_1t + C_2,
\]
where \(C_2\) is another constant of integration.

### Step 2: Apply the initial conditions

#### Initial condition 1: \(x(0) = 1\)
Substitute \(t = 0\) into the expression for \(x(t)\):
\[
x(0) = \frac{0^3}{6} + C_1(0) + C_2 = C_2 = 1.
\]
Thus, \(C_2 = 1\).

#### Initial condition 2: \(\frac{dx(t)}{dt}\bigg|_{t=0} = 2\)
We already have \(\frac{dx(t)}{dt} = \frac{t^2}{2} + C_1\). Substitute \(t = 0\):
\[
\frac{dx(t)}{dt}\bigg|_{t=0} = \frac{0^2}{2} + C_1 = C_1 = 2.
\]
Thus, \(C_1 = 2\).

### Step 3: Final solution

The solution for \(x(t)\) is:
\[
x(t) = \frac{t^3}{6} + 2t + 1.
\]

This matches the first option:

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

Would you like more details or have any questions? 

Here are five related questions:
1. What is the homogeneous solution for this system?
2. How would the solution change if the initial conditions were different?
3. What is the physical meaning of a forced response in this context?
4. Can you solve this problem using Laplace transforms?
5. How does the second derivative in this system affect the shape of the solution?

**Tip**: When solving differential equations, always remember to check if it's homogeneous or non-homogeneous, as that determines the form of the solution.

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

This page provides a step-by-step solution to the differential equation d^2x(t)/dt^2 = t with initial conditions x(0) = 1 and dx(t)/dt|t=0 = 2. Learn how to integrate the equation, apply initial conditions, and find the forced response, suitable for students in Grades 11-12.

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