Solve the initial value problem. dy dx 2 (cos x)- + y sin x = 3x cos ²x, y 5A 6 = 2 The solution is y(x) = cos ²x| 3x sin x + 3 cos x + -49²√√3 48 3√3 2 - 5 4 49²√3 36
Solve the initial value problem. dy dx 2 (cos x)- + y sin x = 3x cos ²x, y 5A 6 = 2 The solution is y(x) = cos ²x| 3x sin x + 3 cos x + -49²√√3 48 3√3 2 - 5 4 49²√3 36
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 Initial Value Problems in Differential Equations
#### Problem Statement:
Solve the initial value problem given by the following differential equation and initial condition:
\[ (\cos x) \frac{dy}{dx} + y \sin x = 3x \cos^2 x, \quad y\left(\frac{5\pi}{6}\right) = \frac{-49\pi^2 \sqrt{3}}{48} \]
#### Solution:
The solution to this initial value problem is given by:
\[ y(x) = \cos^2 (x) \left( 3x \sin x + 3 \cos x + \frac{3\sqrt{3}}{2} - \frac{5\pi}{4} - \frac{49\pi^2 \sqrt{3}}{36} \right) \]
This equation represents the function \( y(x) \) which satisfies the given differential equation and initial condition.
### Explanation and Steps:
1. **Identify the Differential Equation and Initial Condition:**
- The differential equation is:
\[ (\cos x) \frac{dy}{dx} + y \sin x = 3x \cos^2 x \]
- The initial condition is:
\[ y\left(\frac{5\pi}{6}\right) = -\frac{49\pi^2 \sqrt{3}}{48} \]
2. **Solve the Differential Equation:**
- The method to solve this equation typically involves:
- Potential substitution or rearrangement.
- Integrating factor or other relevant techniques from differential equations.
3. **Apply the Initial Condition:**
- Use the initial condition to solve for the constant of integration if necessary.
4. **Construct the Final Solution:**
- Combine the solutions from steps 2 and 3 to form the final expression for \( y(x) \).
### Conclusion:
The function \( y(x) \) provided is a specific solution to the given initial value problem, ensuring that it satisfies both the differential equation and the initial condition specified. Understanding and solving such problems are crucial for applications in various fields including physics, engineering, and applied mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb54c94dc-8fa5-45f1-a08b-382951c33f95%2Fc28037bf-30cf-4c41-b10d-986e83d12c58%2Fbssaxog_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Initial Value Problems in Differential Equations
#### Problem Statement:
Solve the initial value problem given by the following differential equation and initial condition:
\[ (\cos x) \frac{dy}{dx} + y \sin x = 3x \cos^2 x, \quad y\left(\frac{5\pi}{6}\right) = \frac{-49\pi^2 \sqrt{3}}{48} \]
#### Solution:
The solution to this initial value problem is given by:
\[ y(x) = \cos^2 (x) \left( 3x \sin x + 3 \cos x + \frac{3\sqrt{3}}{2} - \frac{5\pi}{4} - \frac{49\pi^2 \sqrt{3}}{36} \right) \]
This equation represents the function \( y(x) \) which satisfies the given differential equation and initial condition.
### Explanation and Steps:
1. **Identify the Differential Equation and Initial Condition:**
- The differential equation is:
\[ (\cos x) \frac{dy}{dx} + y \sin x = 3x \cos^2 x \]
- The initial condition is:
\[ y\left(\frac{5\pi}{6}\right) = -\frac{49\pi^2 \sqrt{3}}{48} \]
2. **Solve the Differential Equation:**
- The method to solve this equation typically involves:
- Potential substitution or rearrangement.
- Integrating factor or other relevant techniques from differential equations.
3. **Apply the Initial Condition:**
- Use the initial condition to solve for the constant of integration if necessary.
4. **Construct the Final Solution:**
- Combine the solutions from steps 2 and 3 to form the final expression for \( y(x) \).
### Conclusion:
The function \( y(x) \) provided is a specific solution to the given initial value problem, ensuring that it satisfies both the differential equation and the initial condition specified. Understanding and solving such problems are crucial for applications in various fields including physics, engineering, and applied mathematics.
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