Question 6 has a solution. Is the solution unique?6. Determine whether or not the initial value problemdyVy; y(0) = 0dxhas a solution. Is the solution unique?7. Consider the differential equationd.xdtkx - x³.se3a. If k≤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 tcritical values c = ±√k are stable.c. Draw the bifurcation diagram-the plot of all points of the form (k, c)is a critical point of the differential equation, with arrows indicating stabilinstability. Note this will be a graph in the k-c plane.

We have given a differential equation, dydx=y3 , y0=0 We know the theorem existence and uniqueness…

Answered: Determine whether or not the initial…

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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Related questions

Question

Question 6

has a solution. Is the solution unique?
6. Determine whether or not the initial value problem
dy
Vy; y(0) = 0
dx
has a solution. Is the solution unique?
7. Consider the differential equation
d.x
dt
kx - x³.
se
3
a. If k≤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 t
critical values c = ±√k are stable.
c. Draw the bifurcation diagram-the plot of all points of the form (k, c)
is a critical point of the differential equation, with arrows indicating stabil
instability. Note this will be a graph in the k-c plane.

Expert Solution

Step 1

We have given a differential equation,

$\frac{d y}{d x} = \sqrt[3]{y}, y (0) = 0$

We know the theorem existence and uniqueness for a general first order differential equation. Given the initial value problem $y' = f (t, y)$ , $y (t_{0}) = y_{0}$ . If $f (t, y)$ is continuous function on rectangle $a < t < b, c < y < d$ and the rectangle contains the point $(t_{0}, y_{0})$ then there exists a solution to the IVP on $(a + h, b - h) \subset (a, b) .$ If additionally the derivative $\frac{\partial f}{\partial y}$ is continuous on the rectangle then the solution is unique.