Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![## Solving the Given Differential Equation
In this section, we will solve the differential equation provided below.
### Differential Equation
\[ 16x^2 y'' + 16xy' - y = 0 \]
### Proposed Solution
\[ y(x) = C_1 \cos \left( \frac{\ln(x)}{4} \right) + C_2 \sin \left( \frac{\ln(x)}{4} \right) \]
where \( \, x > 0 \).
### Explanation
The given differential equation is a second-order linear differential equation with variable coefficients. The solution involves finding the particular form of the function \( y(x) \), which incorporates both cosine and sine functions with a logarithmic argument.
Here:
- \(C_1\) and \(C_2\) are constants determined by initial or boundary conditions.
- \(\ln(x)\) represents the natural logarithm of \(x\).
This is a standard approach to solving such differential equations, where substitution and transformation methods often reduce the problem to a simpler form.
The red cross icon indicates that there may be an error or additional steps needed to verify the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F004f485a-dc8e-4d2a-a997-494cbc4bff99%2F6b0f2b73-7ef3-4752-8480-c55f0bb67b72%2Fnmpku33m_processed.png&w=3840&q=75)
Transcribed Image Text:## Solving the Given Differential Equation
In this section, we will solve the differential equation provided below.
### Differential Equation
\[ 16x^2 y'' + 16xy' - y = 0 \]
### Proposed Solution
\[ y(x) = C_1 \cos \left( \frac{\ln(x)}{4} \right) + C_2 \sin \left( \frac{\ln(x)}{4} \right) \]
where \( \, x > 0 \).
### Explanation
The given differential equation is a second-order linear differential equation with variable coefficients. The solution involves finding the particular form of the function \( y(x) \), which incorporates both cosine and sine functions with a logarithmic argument.
Here:
- \(C_1\) and \(C_2\) are constants determined by initial or boundary conditions.
- \(\ln(x)\) represents the natural logarithm of \(x\).
This is a standard approach to solving such differential equations, where substitution and transformation methods often reduce the problem to a simpler form.
The red cross icon indicates that there may be an error or additional steps needed to verify the solution.
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