# Differential Equations Problem Set

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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#### Solution Preview

(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.

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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