Find the solution to the following initial value problem using Laplace Trans- form: a" + 9x' + 14.x = 20 x(0) = 0, a'(0) = 0.
Find the solution to the following initial value problem using Laplace Trans- form: a" + 9x' + 14.x = 20 x(0) = 0, a'(0) = 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Initial Value Problem Using Laplace Transforms**
**Problem Statement:**
Find the solution to the following initial value problem using Laplace transform:
\[
x'' + 9x' + 14x = 20
\]
with initial conditions:
\[
x(0) = 0, \quad x'(0) = 0.
\]
**Explanation:**
- \(x'' + 9x' + 14x = 20\) represents a second-order linear differential equation where \(x\) is the function of time (or another variable), and the primes denote differentiation with respect to that variable.
- The initial conditions \(x(0) = 0\) and \(x'(0) = 0\) specify the values of the function and its first derivative at the starting point \(t = 0\).
The Laplace transform technique will be used to solve this differential equation by converting it into an algebraic equation, which can be more easily manipulated and solved. Once solved, the inverse Laplace transform will be applied to find the function \(x(t)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8204f301-7e54-4f97-9d0f-3395d2e8f77b%2F6ca51baa-e31f-429b-9d37-575463a30c7c%2Fcefhn3p_processed.png&w=3840&q=75)
Transcribed Image Text:**Initial Value Problem Using Laplace Transforms**
**Problem Statement:**
Find the solution to the following initial value problem using Laplace transform:
\[
x'' + 9x' + 14x = 20
\]
with initial conditions:
\[
x(0) = 0, \quad x'(0) = 0.
\]
**Explanation:**
- \(x'' + 9x' + 14x = 20\) represents a second-order linear differential equation where \(x\) is the function of time (or another variable), and the primes denote differentiation with respect to that variable.
- The initial conditions \(x(0) = 0\) and \(x'(0) = 0\) specify the values of the function and its first derivative at the starting point \(t = 0\).
The Laplace transform technique will be used to solve this differential equation by converting it into an algebraic equation, which can be more easily manipulated and solved. Once solved, the inverse Laplace transform will be applied to find the function \(x(t)\).
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