Math Problem Statement
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
Solution
We are given the second-order differential equation:
with the initial conditions:
Step 1: Solve the differential equation
To solve this, we need to integrate the equation twice to find .
First integration
Integrating both sides with respect to : where is a constant of integration.
Second integration
Now, integrate again to find : where is another constant of integration.
Step 2: Apply the initial conditions
Initial condition 1:
Substitute into the expression for : Thus, .
Initial condition 2:
We already have . Substitute : Thus, .
Step 3: Final solution
The solution for is:
This matches the first option:
Would you like more details or have any questions?
Here are five related questions:
- What is the homogeneous solution for this system?
- How would the solution change if the initial conditions were different?
- What is the physical meaning of a forced response in this context?
- Can you solve this problem using Laplace transforms?
- 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.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Initial Value Problems
Formulas
d^2x(t)/dt^2 = t
dx(t)/dt = ∫t dt = t^2/2 + C₁
x(t) = ∫(t^2/2 + C₁) dt = t^3/6 + C₁t + C₂
Theorems
Integration of functions
Initial Conditions
Suitable Grade Level
Grades 11-12
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