Math Problem Statement
The final answer should be (5s^2+22s+14)/((s+1)^2(s+2))
Solution
To solve this initial value problem using the Laplace transform, we will follow these steps.
The differential equation is: with initial conditions:
Step 1: Take the Laplace Transform of Both Sides
Let . The Laplace transform of derivatives are:
Using the initial conditions and , the Laplace transform of the left side becomes:
Step 2: Expand and Simplify the Left Side
Expanding each term, we get:
Combining terms involving and constants:
Step 3: Take the Laplace Transform of the Right Side
For the right side, , we use the Laplace transform formula:
Step 4: Set Up the Equation and Solve for
Now we have:
Rearrange to solve for :
Step 5: Combine Terms
To make the expression simpler, combine terms over a common denominator:
Expanding the numerator, we get the answer:
Thus,
Would you like further steps for inverse transformation or have questions on any part of the process?
- What is the purpose of using Laplace transforms in differential equations?
- How do initial conditions affect the Laplace transform of a derivative?
- Why is partial fraction decomposition useful in solving for inverse Laplace transforms?
- How would the solution change if initial conditions were different?
- What are common Laplace transforms for basic functions?
Tip: Always check initial conditions carefully when applying the Laplace transform to differential equations. They significantly impact the transformed equation.
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Differential Equations
Initial Value Problems
Formulas
Laplace transform of a derivative: \(\mathcal{L}\{y'\} = sY(s) - y(0)\), \(\mathcal{L}\{y''\} = s^2Y(s) - sy(0) - y'(0)\)
Laplace transform of exponential functions: \(\mathcal{L}\{e^{-at}\} = \frac{1}{s + a}\)
Theorems
Laplace Transform Properties
Linear Differential Equations
Suitable Grade Level
Undergraduate Level
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