Find the solution of the given initial value problem in explicit form. sin(2x) da + cos(9y) dy = 0, y(5) 9 y(x) :
Find the solution of the given initial value problem in explicit form. sin(2x) da + cos(9y) dy = 0, y(5) 9 y(x) :
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:**
Find the solution of the given initial value problem in explicit form.
\[
\sin(2x) \, dx + \cos(9y) \, dy = 0, \quad y \left( \frac{\pi}{2} \right) = \frac{\pi}{9}
\]
**Solution:**
\[
y(x) = \text{[Solution here]}
\]
**Explanation:**
This problem is asking to solve a differential equation with given initial conditions.
1. **Step 1: Separate Variables**
- Rewrite the equation by separating variables involving \( x \) and \( y \).
\[
\sin(2x) \, dx = -\cos(9y) \, dy
\]
2. **Step 2: Integrate Both Sides**
- Integrate both sides of the equation.
\[
\int \sin(2x) \, dx = -\int \cos(9y) \, dy
\]
3. **Step 3: Apply Initial Conditions**
- After finding the integrals, apply the initial condition \( y \left( \frac{\pi}{2} \right) = \frac{\pi}{9} \) to determine the constant of integration.
4. **Step 4: Solve for \( y(x) \)**
- Express the equation in the explicit form \( y = y(x) \).
The box is provided to input the final answer for \( y(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F306db17a-c2a3-4020-aada-7a92dff91485%2F3970f02e-6644-4cf3-badc-b0b8e6dad0ab%2F0jahivh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the solution of the given initial value problem in explicit form.
\[
\sin(2x) \, dx + \cos(9y) \, dy = 0, \quad y \left( \frac{\pi}{2} \right) = \frac{\pi}{9}
\]
**Solution:**
\[
y(x) = \text{[Solution here]}
\]
**Explanation:**
This problem is asking to solve a differential equation with given initial conditions.
1. **Step 1: Separate Variables**
- Rewrite the equation by separating variables involving \( x \) and \( y \).
\[
\sin(2x) \, dx = -\cos(9y) \, dy
\]
2. **Step 2: Integrate Both Sides**
- Integrate both sides of the equation.
\[
\int \sin(2x) \, dx = -\int \cos(9y) \, dy
\]
3. **Step 3: Apply Initial Conditions**
- After finding the integrals, apply the initial condition \( y \left( \frac{\pi}{2} \right) = \frac{\pi}{9} \) to determine the constant of integration.
4. **Step 4: Solve for \( y(x) \)**
- Express the equation in the explicit form \( y = y(x) \).
The box is provided to input the final answer for \( y(x) \).
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