In this problem, x = c₂ cos t + c₂ sint is a two-parameter family of solutions of the second-order DE x" + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. x(x/3) = V x'(x/3) = 0 Need Help? Read It Watch It
In this problem, x = c₂ cos t + c₂ sint is a two-parameter family of solutions of the second-order DE x" + x = 0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. x(x/3) = V x'(x/3) = 0 Need Help? Read It Watch It
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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![### Solving Second-Order Differential Equations: Initial Value Problem
In this problem, \( x = c_1 \cos t + c_2 \sin t \) is a two-parameter family of solutions of the second-order differential equation \( x'' + x = 0 \).
**Objective:** Find a solution of the second-order initial value problem (IVP) consisting of this differential equation and the given initial conditions.
#### Given Conditions:
\[ x\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
\[ x'\left(\frac{\pi}{3}\right) = 0 \]
#### Solution:
To find the specific solution \( x \), substitute the given conditions into the general solution and solve for the constants \( c_1 \) and \( c_2 \).
\[ x = \underline{\phantom{C}} \]
Need Help?
- [Read It](#)
- [Watch It](#)
#### Explanation:
The given differential equation is homogeneous, and we can solve it using the characteristic equation or by applying known trigonometric solutions due to its simple harmonic motion nature.
For further assistance, utilize the "Read It" and "Watch It" resources provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0436118d-47b7-4fa9-abd3-dac72bbeccd1%2F4488c772-1fdb-4194-b9d5-5ffdad1f58d6%2Fhvs6k4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Second-Order Differential Equations: Initial Value Problem
In this problem, \( x = c_1 \cos t + c_2 \sin t \) is a two-parameter family of solutions of the second-order differential equation \( x'' + x = 0 \).
**Objective:** Find a solution of the second-order initial value problem (IVP) consisting of this differential equation and the given initial conditions.
#### Given Conditions:
\[ x\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
\[ x'\left(\frac{\pi}{3}\right) = 0 \]
#### Solution:
To find the specific solution \( x \), substitute the given conditions into the general solution and solve for the constants \( c_1 \) and \( c_2 \).
\[ x = \underline{\phantom{C}} \]
Need Help?
- [Read It](#)
- [Watch It](#)
#### Explanation:
The given differential equation is homogeneous, and we can solve it using the characteristic equation or by applying known trigonometric solutions due to its simple harmonic motion nature.
For further assistance, utilize the "Read It" and "Watch It" resources provided.
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