dy dt 7=2 €²-1 b) y (₁)=2 b) y(-1)=2 (c) y(₁) =3 d) y (0) = 2 Seproble method e) stote, which for initial conditions. the Particular solution exist and for which it does not exist
dy dt 7=2 €²-1 b) y (₁)=2 b) y(-1)=2 (c) y(₁) =3 d) y (0) = 2 Seproble method e) stote, which for initial conditions. the Particular solution exist and for which it does not exist
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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![The image contains handwritten notes related to solving a differential equation. The transcription of the text along with a detailed explanation is provided below:
---
### Differential Equations: Separable Method
Given the differential equation:
\[ \frac{dy}{dt} = \frac{y - 2}{t^2 - 1} \]
**Initial Conditions for Particular Solutions:**
a) \( y(1) = 2 \)
b) \( y(-1) = 2 \)
c) \( y(1) = 3 \)
d) \( y(0) = 2 \)
**Task:**
e) State which initial conditions the particular solution exists for and for which it does not exist.
---
### Detailed Explanation:
This text involves solving a first-order differential equation of the form \(\frac{dy}{dt} = \frac{y - 2}{t^2 - 1} \), using the separable method which involves separating the variables \(y\) and \(t\) on different sides of the equation and then integrating both sides.
The initial conditions provided in items (a) to (d) are specific values for \(y\) at certain \(t\)-values. To determine whether a particular solution exists for these initial conditions, one needs to analyze if the differential equation and its separation are valid at those points, particularly checking for any singularities or undefined expressions in the domain of the solution.
For example, a potential issue arises when \( t = \pm1 \) because the denominator \( t^2 - 1 \) becomes zero, which leads to a division by zero scenario. Therefore, it is necessary to check these points against the initial conditions to conclude the existence and uniqueness of the solution.
---
This detailed explanation helps students understand the process of solving the differential equation and determining the existence of solutions based on given initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0feac75-9cf9-420d-995e-ce89c47a4906%2F8b0dbb1d-5b7f-4374-b9db-f15a5cea4777%2Fmtlw76n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains handwritten notes related to solving a differential equation. The transcription of the text along with a detailed explanation is provided below:
---
### Differential Equations: Separable Method
Given the differential equation:
\[ \frac{dy}{dt} = \frac{y - 2}{t^2 - 1} \]
**Initial Conditions for Particular Solutions:**
a) \( y(1) = 2 \)
b) \( y(-1) = 2 \)
c) \( y(1) = 3 \)
d) \( y(0) = 2 \)
**Task:**
e) State which initial conditions the particular solution exists for and for which it does not exist.
---
### Detailed Explanation:
This text involves solving a first-order differential equation of the form \(\frac{dy}{dt} = \frac{y - 2}{t^2 - 1} \), using the separable method which involves separating the variables \(y\) and \(t\) on different sides of the equation and then integrating both sides.
The initial conditions provided in items (a) to (d) are specific values for \(y\) at certain \(t\)-values. To determine whether a particular solution exists for these initial conditions, one needs to analyze if the differential equation and its separation are valid at those points, particularly checking for any singularities or undefined expressions in the domain of the solution.
For example, a potential issue arises when \( t = \pm1 \) because the denominator \( t^2 - 1 \) becomes zero, which leads to a division by zero scenario. Therefore, it is necessary to check these points against the initial conditions to conclude the existence and uniqueness of the solution.
---
This detailed explanation helps students understand the process of solving the differential equation and determining the existence of solutions based on given initial conditions.
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