Question

Consider the autonomous first-order differential equation dy/dx=4y-(y^3).

1. Classify each critical point as asymptotically stable, unstable, or semi-stable. (DO NOT draw the phase portrait and DO NOT sketch the solution curves)

2. Solve the Bernoulli differential equation dy/dx=4y-(y^3).

Answer #1

] Consider the autonomous differential equation y 0 = 10 + 3y −
y 2 . Sketch a graph of f(y) by hand and use it to draw a phase
line. Classify each equilibrium point as either unstable or
asymptotically stable. The equilibrium solutions divide the ty
plane into regions. Sketch at least one solution trajectory in each
region.

Consider the autonomous differential equation dy/ dt = f(y) and
suppose that y1 is a critical point, i.e., f(y1) = 0.
Show that the constant equilibrium solution y = y1 is
asymptotically stable if f 0 (y1) < 0 and unstable if f 0 (y1)
> 0.

For
the autonomous differential equation dy/dt=1-y^2, sketch a graph of
f(y) versus y, identify the equilibrium solutions identify them as
stable, semistable or unstable, draw the phase line and sketch
several graphs of solutions in the ty-plane.

Given dy/dx =y(y−3)(1−y)^2 dx (a) Sketch the phase line
(portrait) and classify all of the critical (equilibrium) points.
Use arrows to indicated the flow on the phase line (away or towards
a critical point). (b) Next to your phase line, sketch the graph of
solutions satisfying the initial conditions: y(0)=0, y(0)=1,
y(0)=2, y(0)=3, y(2)=4, y(0)=5.

Use the method for solving Bernoulli equations to solve the
following differential equation.
(dy/dx)+4y=( (e^(x))*(y^(-2)) )
Ignoring lost solutions, if any, the general solution y=
______(answer)__________
(Type an expression using x as the variable)
THIS PROBLEM IS A DIFFERENTIAL EQUATIONS PROBLEM. Only people
proficient in differential equations should attempt to solve.
Please write clearly and legibly. I must be able to CLEARLY read
your solution and final answer. CIRCLE YOUR FINAL ANSWER.

Solve the given differential equation
y-x(dy/dx)=3-2x2(dy/dx)

1. Sketch the direction field for the following differential
equation dy dx = y − x. You may use maple and attach your graph.
Also sketch the solution curves with initial conditions y(0) = −1
and y(0) = 1.

Solve the first order homogeneous differential equation:
(2x-5y)dx + (4x-y) dy=0

differential equations solve
(2xy+6x)dx+(x^2+4y^3)dy, y(0)=1

3. Consider the differential equation: x dy/dx = y^2 − y
(a) Find all solutions to the differential equation.
(b) Find the solution that contains the point (−1,1)
(c) Find the solution that contains the point (−2,0)
(d) Find the solution that contains the point (1/2,1/2)
(e) Find the solution that contains the point (2,1/4)

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