3. Solve the given initial value problem. Determine the period and sketch the graph of the solution. (a) x² + 4x = 0, (b) x" +16x = 0, x(0) = 2, x (0) = 2, x' (0) = 1. x'(0) = 1. no
3. Solve the given initial value problem. Determine the period and sketch the graph of the solution. (a) x² + 4x = 0, (b) x" +16x = 0, x(0) = 2, x (0) = 2, x' (0) = 1. x'(0) = 1. no
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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Question
Transcribed Image Text:**3. Solve the given initial value problem. Determine the period and sketch the graph of the solution.**
(a) \( x'' + 4x = 0, \quad x(0) = 2, \quad x'(0) = 1. \)
(b) \( x'' + 16x = 0, \quad x(0) = 2, \quad x'(0) = 1. \)
**Explanation:**
1. **Problem Overview:**
- You are required to solve two differential equations with given initial conditions.
- After solving each differential equation, determine the period of the solutions.
- Sketch the graph of the solutions.
2. **Mathematical Approach:**
- Recognize that each problem is a second-order linear homogeneous differential equation.
- Use characteristic equations to solve for the general solution of each differential equation.
- Apply initial conditions to find specific solutions for each initial value problem.
3. **Graph Explanation:**
- The graphs you sketch will show oscillatory motion typical of solutions to such differential equations.
- The period is related to the coefficients of the differential equations and the characteristics of trigonometric functions incorporated into solutions.
These tasks involve mathematical understanding of differential equations, initial value problems, and periodic functions.
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