ï + x + ex = 0, with x(0) = 1, ¿(0) = 1.
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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Using pertubation theory, find x0, x1, and x2 in the series expansion x(t,epsilon)=x0(t)+epsilon x1(t) + epsilon^2 x2(t) + O(epsilon^3).
![This differential equation is given by:
\[
\ddot{x} + x + \epsilon x = 0,
\]
with initial conditions:
\[
x(0) = 1, \quad \dot{x}(0) = 1.
\]
### Explanation:
This is a second-order linear differential equation with a small parameter \(\epsilon\). The function \(x(t)\) represents the unknown solution, with its second derivative given by \(\ddot{x}\) and first derivative by \(\dot{x}\).
### Initial Conditions:
- \(x(0) = 1\): the initial position at time \(t = 0\) is 1.
- \(\dot{x}(0) = 1\): the initial velocity at time \(t = 0\) is 1.
This equation might be used to model different physical phenomena, like oscillations in mechanical or electrical systems, depending on the value of \(\epsilon\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F26f4eae0-3c49-4cd6-9bb2-65a0cd080993%2F280j20j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This differential equation is given by:
\[
\ddot{x} + x + \epsilon x = 0,
\]
with initial conditions:
\[
x(0) = 1, \quad \dot{x}(0) = 1.
\]
### Explanation:
This is a second-order linear differential equation with a small parameter \(\epsilon\). The function \(x(t)\) represents the unknown solution, with its second derivative given by \(\ddot{x}\) and first derivative by \(\dot{x}\).
### Initial Conditions:
- \(x(0) = 1\): the initial position at time \(t = 0\) is 1.
- \(\dot{x}(0) = 1\): the initial velocity at time \(t = 0\) is 1.
This equation might be used to model different physical phenomena, like oscillations in mechanical or electrical systems, depending on the value of \(\epsilon\).
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