Question

**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.**

Answer #1

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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.

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.

following nonlinear system:
x' = 2 sin y,
y'= x^2 + 2y − 1
find all singular points in the domain x, y ∈ [−1, 1],determine
their types and stability.
Find slopes of saddle separatrices.
Use this to sketch the phase portrait in the domain x, y ∈ [−1,
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 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).

a)
find all possible solutions of y''+y'-6y=12t
b) solve initial value problem of y''+y'-6y=12t, y(0)=1,
y'(0)=0

Solve the following system using augmented matrux methods
-3x+6y = 0
-4x +6y = 0
a) The initial matrix is:
b) First, perform the Row Operation 1/-3R1->R1. The resulting
matrix is:
c) Next, perform the operation +3R1+R2->R2. The resulting
matrix is:
d) Finish simplifying the augmented matrix. The reduced matrix
is:
e) How many solutions does the system have?
f) What are the solutions to the system?
x =
y =

1) Solve the system of equations. Give your answer as an
ordered pair (x,y)
{y=−7
{5x-6y=72
a) One solution:
b) No solution
c) Infinite number of solutions
2) Solve the system of equations. Give your answer as an
ordered pair (x,y)
{x=2
{3x-6y=-30
a) One solution:
b) No solution
c) Infinite number of solutions

find the general solution.
1- y^6(4)+12y''+36y=0
2-6y^(4)+5y'''+7y''+5y'+y=0
3-y^(4)-4y'''+7y''-6y'+2y=0

Differential Equations. Solve the following IVP. Y''(Double
Prime) + 6Y'(Prime)+5Y =0, Y'(Prime)(0) =0, Y(0)=1

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