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2) Sole the Rollowing liníkal value problems. for each problem, determine whether the systen is over-, or critically damped. non inder-, a) 4y"+9y! +2y=0 6) y" + by! + loy=o c) yu + 4y! t 3y=0

2) Sole the Rollowing liníkal value problems. for each problem, determine whether the systen is over-, or critically damped. non inder-, a) 4y"+9y! +2y=0 6) y" + by! + loy=o c) yu + 4y! t 3y=0

Advanced Engineering Mathematics
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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**Problem Statement:** Solve the following initial value problems. For each problem, determine whether the system is underdamped, overdamped, or critically damped. a) \( 4y'' + 9y' + 2y = 0 \) b) \( y'' + 6y' + 10y = 0 \) c) \( y'' + 4y' + 3y = 0 \) **Instructions:** 1. Identify the type of damping for each differential equation by analyzing the characteristic equation. 2. Solve the differential equations to find the general solutions. 3. Use the determined damping type to describe the system behavior. **Note:** - "Underdamped" systems have complex conjugate roots with negative real parts. - "Overdamped" systems have distinct real roots. - "Critically damped" systems have repeated real roots.
Transcribed Image Text:**Problem Statement:** Solve the following initial value problems. For each problem, determine whether the system is underdamped, overdamped, or critically damped. a) \( 4y'' + 9y' + 2y = 0 \) b) \( y'' + 6y' + 10y = 0 \) c) \( y'' + 4y' + 3y = 0 \) **Instructions:** 1. Identify the type of damping for each differential equation by analyzing the characteristic equation. 2. Solve the differential equations to find the general solutions. 3. Use the determined damping type to describe the system behavior. **Note:** - "Underdamped" systems have complex conjugate roots with negative real parts. - "Overdamped" systems have distinct real roots. - "Critically damped" systems have repeated real roots.
Expert Solution
Step 1

To find- Solve the following non initial value problems. For each problem, determine whether the system is under-, over-, or critically damped.

  1. 4y'' + 9y' + 2y = 0
  2. y'' + 6y' + 10y = 0
  3. y'' + 4y' + 3y = 0

 

Concept Used- 

If the equation is of the form ay'' + by' + cy = 0 where a > 0, b ≥ 0, c > 0

The Characteristic equation can be written it as ar2 + br + c = 0 with characteristic roots

-b ± b2 - 4ac2a

Then there are three cases possible on the basis of sign under the square root

  1. Case I: b2 < 4ac {It will be the under-damping case}
  2. Case II: b2 > 4ac {It will be the over-damping case}
  3. Case I: b2 = 4ac {It will be the critical-damping case}
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