Exercise MA 282-1Consider the initial value problem: t y '+3 y=5 t 2, y (2 )=5

a. Solve using dsolveb. b. Plot the solution y (t ) on the intervals 0.5≤ t ≤5 and 0.2≤t ≤20. Determine the behavior of y (t ) as t approaches 0 from the right and as t becomes large.c. Change the initial condition to y (2 )=3 and determine the behaviour of this solution, again by plotting on intervals such as 0.2≤t ≤20 and 0.5≤ t ≤5.Partial Solution

a. In the command window, type:

b. Plot of the solution on the interval 0.2≤t ≤20

c. Plot of the solution on the interval 0.2≤t ≤20

Exercise MA 282-2a. Solve the initial value problem: y '−2 y=sin 2 t , y (0 )=cb. Use MATLAB to graph solutions for c=−0.5 ,−0.45 ,…,−0.05 ,0 on the interval [0, 1]. Use axis tight.c. Use MATLAB to graph solutions for c=−0.5 ,−0.45 ,…,−0.05 ,0 on the interval [0, 2.5]d. What happens to the solution curves as t increases?e. What effect small changes in initial data can have on the global behavior of solutions curves?

Partial Solutiona. In the command window type:

b. Make sure you use "axis tight"

Exercise MA282-3Consider the critical threshold model for population growth y '=−(2− y ) y.

a. Find the equilibrium solutions of the differential equation.b. Draw the slop field and use it to decide which equilibrium solutions are stable and which are unstable (y (t ) is stable if solutions that start near y (t ) converge to y (t )). Hint: modify [T,Y]=meshgrid(-4:0.2:4, -4:0.2:4)until you obtain a desirable slope field.c. What is the limiting behavior of the solution if the initial population is between 0 and 2? Greater than 2?d. Use dsolve to find the solutions with initial values 1.5, 0.3, and 2.1. e. Plot the three solutions of part (d) together with the slop field on the same graph. Do the solutions follow the slope field as you expect it to?

Partial solutionPart (b)

Part (d)

Exercise MA282-4Consider y '=(α−1 ) y− y3.

a. Use solve to find the roots of (α−1 ) y− y3. Explain why y=0 is the only real root when α≤1, and why there are three distinct roots when α>1.b. For α=−2 ,−1,∧2, draw a direction field for the differential equation and deduce there is only one equilibrium solution. Is it stable?c. Do the same for α=1.d. For α=1.5 ,2 ,draw the direction field . Identify the equilibrium solutions and explain their stability.e. Explain: "As α increase through 1, the stable solution x=0 bifurcates into two stable solutions"

Exercise MA282-5Consider the differential equation y '= t

y .a. Use ode45 to calculate and plot an approximate solution to the initial value problem: y '= ty , y (0 )=1 for 0≤ t ≤2b. Use dsolve to find an exact solution of the IVPc. Plot the exact and the approximate solution on the same graph. can yo distinguish between the two curves?d. Plot a family of solutions with initial values y (0 )=1.2 ,1.4 ,1.6 ,1.8 over the interval [-1, 2]

Partial Solutiona. For details, see the notes

b.

c.Exercise MA282-6Consider the predator-prey system {dRdt =2(1− R3 )R−RF

¿

dFdt

=−2 F+4 RF . (See Blanchard section 2.1)

a. Use solve to find all the equilibrium solutionsb. Plot the solution curve corresponding to the initial conditions R (1 )=1 ,∧F (1 )=3.7 over the interval 0≤ t ≤30. For axis, use [0, 2, 0, 6].

c. Describe the fate of the prey and predator populations based on the solution curve.d. Display graphs of the solutions R(t ) and F (t) corresponding to the initial conditions R (1 )=1 ,∧F (1 )=3.7 over the interval 0≤ t ≤30. Compare with part (c). Solution

a. There are three equilibrium solutions:

b. Make sure you solve forwards and backwards:

c. Describe the fate of the prey and predator populations based on the solution curve.d. A loop reduces the amount of typing:

Exercise MA282-7 (Blanchard)a. Convert the second-order differential equation d2 x

d t 2+2 dxdt

−3 x+x3=0 into a first-order system in terms of x and v, where v=dxdt b. Find all equilibrium pointsc. Use MATLAB to sketch the associated direction field. Hint: use meshgrid(-5:.4:5,-5:.4:5)and axis tightd. The following script plots several solution curves over the interval −3≤ t ≤3

What are the initial conditions?e. Use the above script to plot a phase portrait together with the direction field of part c.f. Describe the behavior of the solutions?

Solutiona. The first-order system is { dx

dt=v

dvdt

=3 x−x3−2v

b. We use the command solve to find the equilibrium points:

f=@(t,P) [P(2);3*P(1)-P(1)^3-2*P(2)];a=-2:0.2:2;b=-2:0.2:2;for k=1:length(a) [t,Pap]=ode45(f, [0,3],[a(k),b(k)]);plot(Pap(:,1),Pap(:,2),'linewidth',2)hold on[t,Pap]=ode45(f, [0,-3],[a(k),b(k)]);plot(Pap(:,1),Pap(:,2),'linewidth',2)endhold offaxis([-5,5,-5,5])

c. Be careful how to use the V

d. (x (0 ) , v (0 ) )=(a (k ) , b (k ))

e.

Exercise MA282-8Let a be a real number different from 0 and consider the system

dxdt

=ax+4 y

dydt=ay

a. Use MATLAB to verify that the system has a double eigenvalue λb. Use MATLAB to verify that there is only one eigenvector associated with λc. Write a program that calculates the general solution of the systemd. Verify your program by using dsolve to solve the systemSolution

Exercise MA282-9 (Blanchard Ex 19, 3.3)Consider the initial value problem

dxdt

=−2x+ 12y

dydt =− y

, x (0 )=m, y (0 )=n.

1. Draw a direction field for the system. 2. Determine the type of the equilibrium point at the origin.3. Use dsolve to solve the IVP in terms of m and n4. Find all straight-line solutions5. Plot the straight-line solutions together with the solutions with initial conditions (m ,n )=(2 ,1 ) , (1 ,−2 ) , (−2 ,2 ) ,(−2,0)