Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. (t – 4)y + (Int)y = 6t, y(1) = 6 i
Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. (t – 4)y + (Int)y = 6t, y(1) = 6 i
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Determining the Interval of Solution Existence for an Initial Value Problem
**Problem Statement:**
Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist.
**Given Equation:**
\[ (t - 4)y' + (\ln{t})y = 6t, \quad y(1) = 6 \]
**Your Task:**
Identify the interval for \( t \) where the solution is guaranteed. The equation given is a differential equation with an initial value. To find this interval, consider the domain in which the functions involved are continuous and the coefficients are properly defined.
**Interval Representation:**
\[ \boxed{\phantom{a}} < t < \boxed{\phantom{a}} \]
### Explanation of Provided Diagram:
The provided diagram includes two blank boxes with information icons, where you need to input the appropriate interval bounds. These bounds indicate the range for the variable \( t \).
To determine the interval, analyze the differential equation and identify the constraints:
- \((t - 4)\): This term tells us there might be a problem when \( t = 4 \) because the coefficient would become zero.
- \(\ln{t}\): The natural logarithm function \(\ln{t}\) is only defined for \( t > 0 \).
Given these constraints, consider \( t \) within the bounds where the equation remains valid and the initial condition \( y(1) = 6 \) is meaningful.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77cac1a6-5ad9-4f72-bdd9-21a202e53df4%2F6d5bca34-b58c-4fc5-8bef-187c23d7e89f%2Fpjuvia_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining the Interval of Solution Existence for an Initial Value Problem
**Problem Statement:**
Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist.
**Given Equation:**
\[ (t - 4)y' + (\ln{t})y = 6t, \quad y(1) = 6 \]
**Your Task:**
Identify the interval for \( t \) where the solution is guaranteed. The equation given is a differential equation with an initial value. To find this interval, consider the domain in which the functions involved are continuous and the coefficients are properly defined.
**Interval Representation:**
\[ \boxed{\phantom{a}} < t < \boxed{\phantom{a}} \]
### Explanation of Provided Diagram:
The provided diagram includes two blank boxes with information icons, where you need to input the appropriate interval bounds. These bounds indicate the range for the variable \( t \).
To determine the interval, analyze the differential equation and identify the constraints:
- \((t - 4)\): This term tells us there might be a problem when \( t = 4 \) because the coefficient would become zero.
- \(\ln{t}\): The natural logarithm function \(\ln{t}\) is only defined for \( t > 0 \).
Given these constraints, consider \( t \) within the bounds where the equation remains valid and the initial condition \( y(1) = 6 \) is meaningful.
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