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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.

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

#### 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.

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