5. Determine whether or not the initial value problem dy dx has a solution. Is the solution unique? √x-y; y(2) = 1
5. Determine whether or not the initial value problem dy dx has a solution. Is the solution unique? √x-y; y(2) = 1
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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Question 5
![**Educational Website Content: Differential Equations**
**Problem 5: Initial Value Problem**
Determine whether the initial value problem
\[
\frac{dy}{dx} = \sqrt{x - y}; \quad y(2) = 1
\]
has a solution. Is the solution unique?
**Problem 6: Initial Value Problem**
Determine whether the initial value problem
\[
\frac{dy}{dx} = \sqrt[3]{y}; \quad y(0) = 0
\]
has a solution. Is the solution unique?
**Problem 7: Differential Equation Analysis**
Consider the differential equation
\[
\frac{dx}{dt} = kx - x^3.
\]
- a. If \( k \leq 0 \), show that the only critical value \( c = 0 \) of \( x \) is stable.
- b. If \( k > 0 \), show that the critical value \( c = 0 \) of \( x \) is now unstable, but that the critical values \( c = \pm \sqrt{k} \) are stable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F590efbb4-bfa3-48e4-b7a4-ba8d0f00ec23%2F7729e889-04b4-4e34-aca6-61078dcd0b51%2Fqkt5zj3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content: Differential Equations**
**Problem 5: Initial Value Problem**
Determine whether the initial value problem
\[
\frac{dy}{dx} = \sqrt{x - y}; \quad y(2) = 1
\]
has a solution. Is the solution unique?
**Problem 6: Initial Value Problem**
Determine whether the initial value problem
\[
\frac{dy}{dx} = \sqrt[3]{y}; \quad y(0) = 0
\]
has a solution. Is the solution unique?
**Problem 7: Differential Equation Analysis**
Consider the differential equation
\[
\frac{dx}{dt} = kx - x^3.
\]
- a. If \( k \leq 0 \), show that the only critical value \( c = 0 \) of \( x \) is stable.
- b. If \( k > 0 \), show that the critical value \( c = 0 \) of \( x \) is now unstable, but that the critical values \( c = \pm \sqrt{k} \) are stable.
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