Given dy/dx = y^2 − 4y + 4 (a) Sketch the phase line (portrait) and classify...

Given dy/dx = y^2 − 4y + 4

(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(0)=4.

(c) Find lim y(x) x→∞ for the solution satisfying the inital condition y(0) = 2.

(d) State the solution to the initial-value problem dy/dx = y^2 − 4y + 4, y(0) = 2.

Expert Solution

If you have any doubts in the solution please ask me in comments...

In first part i draw phase line

Second part i draw direction field then i draw solution curve with help of direction field

Third we simply use curve for limit

In 4 we find y=2 is singular slolution

Colby Messinger answered 1 year ago

Sketch the direction field of the equation dy/dx=y-4y^3. Sketch the phase portrait. Find the equilibrium solutions...

Sketch the direction field of the equation dy/dx=y-4y^3. Sketch the phase portrait. Find the equilibrium solutions and classify each equilibrium as stable, unstable or semi-stable. Sketch typical solutions of the equation.

Consider the differential equation dy/dx = y^2 + y - 2 (1) Sketch its phase portrait...

Consider the differential equation dy/dx = y^2 + y - 2 (1) Sketch its phase portrait and classify the critical points. (2) Find the explicit solution of the DE.

d^2y/dx^2 − dy/dx − 3/4 y = 0, y(0) = 1, dy/dx(0) = 0, Convert the...

d^2y/dx^2 − dy/dx − 3/4 y = 0, y(0) = 1, dy/dx(0) = 0, Convert the initial value problem into a set of two coupled first-order initial value problems and find the exact solution to the differential equatiion

Find the general solution of the given system. dx/dt=6x-y dy/dt=5x+4y

a.Express y in terms of x given that dy/dx = (y + 2)(2x + 1) given...

a.Express y in terms of x given that dy/dx = (y + 2)(2x + 1) given that y = 2 at x = 0. b. Solve (x^2 + 1)dy/dx + 3xy = 6x. c) Obtain a general solution of dy/dx + y/x = sin x.

Solve these first-order Differential Equations using an integrating factor. 1. dy/dx+2xy=0 2. dy/dx-y=5 3. dy/dx+y=x 4....

Solve these first-order Differential Equations using an integrating factor. 1. dy/dx+2xy=0 2. dy/dx-y=5 3. dy/dx+y=x 4. (x)dy/dx+(x+1)y=3/x 5. (x^2)dy/dx=e^x-2xy

1. (a) Sketch the slope field for the given differential equation: dy/dx = 2? (b) Find...

1. (a) Sketch the slope field for the given differential equation: dy/dx = 2? (b) Find the particular solution of the differential equation that satisfies the initial condition y(0) = 4 (c) What is the value of y when x = 1/2 2. (a) Find the general solution of the given differential equation: dy/dx = ysinx = ????? 2 (b) Find the particular solution of the differential equation that satisfies the initial condition ? = 2; ?ℎ?? ? = π/2

Solve the given initial-value problem. (x + y)2 dx + (2xy + x2 − 2) dy...

Solve the given initial-value problem. (x + y)2 dx + (2xy + x2 − 2) dy = 0, y(1) = 1

dy/dx + y/x \ x3y2

Let Q1=y(2) Q2 =y(3) where y=y(x) solves y'+2xy=2x^3 y(0)=1 dy/dx= (2x-y)/(x+4y) y(1)=1 y'+ycotx=y^3sin^3x y(pi/2)=1 (y^2-2x)dx+2xy dy=0 ...

Let Q1=y(2) Q2 =y(3) where y=y(x) solves y'+2xy=2x^3 y(0)=1 dy/dx= (2x-y)/(x+4y) y(1)=1 y'+ycotx=y^3sin^3x y(pi/2)=1 (y^2-2x)dx+2xy dy=0 y(1)=1 2x^2y;+4xy=3sinx y(2pi)=0 y"+2y^-1 (y')^2=y'

Question