Question

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.

Answer #1

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).

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.

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.

Sketch the phase portrait of y” - y’ - 6y = 0, y(0) = 2,
y’(0) = 3 as an autonomous system of two first order equations and
discuss the stability and the long time behavior of the
solutions.

Consider the equation:
x'=x^3-3x^2+2x
sketch the phase line. solve the equation and sketch the graphs
of some solutions including at least one solution with values in
each interval above, below and between the critical points.
identify critical points as stable or unstable

Consider the initial value problem
dy dx
=
1−2x 2y
, y(0) = − √2
(a) (6 points) Find the explicit solution to the initial value
problem.
(b) (3 points) Determine the interval in which the solution is
deﬁned.

Find a solution of x dy dx = y2 − y that passes through the
indicated points. (a) (0, 1) y =
(b) (0, 0) y =
(c) 1/ 3 , 1 /3 y =
(d) 2, 1/ 8 y =

Solve: 1.dy/dx=(e^(y-x)).secy.(1+x^2),y(0)=0.
2.dy/dx=(1-x-y)/(x+y),y(0)=2 .

Use Implicit Differentiation to find first dy/dx , then the
equation of the tangent line to the curve x2+xy+y2= 2-y at the
point (0,-2)
b. Determine a function of the form f(x)= ax2+ bx + c (that is,
find the real numbers a,b,c ) if the graph of the function has
slope 2 at the point (3,4) , and has a horizontal tangent where
x=1
c. Assume that x,y are functions of variable t satisfying the
equation x2+xy=10. Find dy/dt...

1.Given that y = x + tan−1 y , find dy dx
2.Determine the equation of the tangent line to the curve y = (2
+ x) e −x at the point (0, 2)

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