Math Problem Statement
Solution
Let’s break this question paper into solutions for each problem step by step.
QUESTION TWO
Part (a): Laplace Transform
-
(i)
- To find the inverse Laplace transform of a rational function, we typically factorize the denominator .
- Factorize:
- Rewrite the function using partial fraction decomposition: Multiply through: Expand and solve for and .
-
(ii)
- Factorize the denominator:
- Apply partial fraction decomposition similarly: Solve for constants and to complete.
Part (b): Solve Differential Equation Using Laplace Transform
The given equation is: Steps to solve:
-
Take the Laplace Transform of both sides:
- Recall properties:
-
Substitute into the equation: Simplify and solve for .
-
Apply the Inverse Laplace Transform to find .
QUESTION THREE
Part (b): Define Periodic Function
The given periodic function has a repeating triangular waveform with peaks and zero crossings:
- Period (based on the graph).
- Define analytically using a piecewise function:
- Extend periodically with .
Part (c): Solve Differential Equation with Laplace Transform
The equation is: Steps to solve:
- Take the Laplace Transform of both sides:
- Use:
- Substitute initial conditions and .
- Solve for (Laplace domain).
- Apply the Inverse Laplace Transform to find .
Questions for Deeper Understanding:
- How do we factorize higher-order polynomials when solving Laplace transforms?
- What is the importance of initial conditions in solving differential equations?
- Can you explain the periodic function using Fourier series representation?
- How do Laplace transforms handle non-homogeneous equations?
- What happens if the periodic function has a non-zero average value?
Tip:
When using Laplace Transforms to solve ODEs, always remember to carefully apply initial conditions after transforming the equation.
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Differential Equations
Partial Fractions
Periodic Functions
Formulas
Laplace Transform of derivative: L{y'} = sY(s) - y(0)
Inverse Laplace Transform
Partial Fraction Decomposition for rational functions
Theorems
Laplace Transform properties
Superposition Principle for solving differential equations
Suitable Grade Level
University Level
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