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Differential Equations Problem Set

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Consider the differential equation

dy/dx = -x/y.

a) Sketch a direction field for this differential equation.
b) Sketch solution curves of the equation passing through the points (0, 1), (1, 1) and (0, -2).
c) State the regions of the xy-plane in which the conditions of the existence and uniqueness theorem are satisfied. Quote the theorem which you are using.
d) Consider the autonomous equation

dx/dt = -3/2 * third root of x.

Show that the trajectory passing through a point x_0 > 0 reaches a fixed point in finite time. Explain why it is possible in this case to have more than one trajectory passing through a point of the phase space.

Please refer to the attached document for the complete problem set.

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https://brainmass.com/math/calculus-and-analysis/differential-equations-problem-set-520401

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(a) We wish to sketch a direction field for this differential equation.

We do so by drawing the vector (y, -x) at each point (x, y). The result is shown ...

Solution Summary

This solution shows step-by-step calculations to solve various problems involving ordinary differential equations. Explanations are also included for understanding.

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See Also This Related BrainMass Solution

Lipschitz continuity and its role in the existence and uniqueness of ordinary differential equations is investigated.

PROBLEM 1. Find a Lipschitz constant, K, for the function f (u, t) = u^3 + t u^2 which shows that f is Lipschitz in u on the set 0 ? u ? 2, 0 ? t ? 1.

PROBLEM 2. Show that the function f (u, t) = t u^(1/2), is not Lipschitz in u on [0, 1] × [0, 2].

PROBLEM 3. Find two solutions to the initial value problem y = |y|^(1/2) , y(0) = 0. What hypothesis of the Picard-Lindelöf Theorem is violated?

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