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.© BrainMass Inc. brainmass.com March 5, 2021, 12:36 am ad1c9bdddf
(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 ...
This solution shows step-by-step calculations to solve various problems involving ordinary differential equations. Explanations are also included for understanding.