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Chapter 9: Q45E (page 442)

Use theorem 9.3.13 to solve the initial value problem

dx鈫�dt=(1201)x鈫�withx鈫�(0)=[1-1] .

Hint: Find first x2(t) and then x1(t).

Short Answer

Expert verified

The solution iset1-2t-1

Step by step solution

01

Definition of first order linear differential equation.

Consider the differential equation f'(t)-af(t)=g(t)where g(t)is a smooth function and 'a'is a constant. Then the general solution will bef(t)=eat鈭�e-atg(t)dt.

02

Determination of the solution.

Consider the differential equation as follows.

dx鈫�dt=1201x鈫�A=1201A=1f't-1ft=0

Now, the differential equation is in the form as follows.

f't-aft=gt, where g(t) is a smooth function, then the general solution will be as follows.

ft=eat鈭�e-atgtdt

03

Compute the calculation of the solution.

Substitute the value 0 for g(t) and 1 for a inft=eat鈭�e-atgtdtas follows.

ft=eat鈭�e-atgtdtft=et鈭�e-t脳0脳dtft=et鈭�0脳dtft=et.C

Here C is constant

Also it can be written as follows.

C=1-2t-1

The solution iset1-2t-1

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Most popular questions from this chapter

Consider an n脳n matrix A with m distinct eigenvalues 位1,位2,鈥�,位m.

(a) Show that the initial value problemdx鈫�dt=Ax鈫� withrole="math" localid="1660807946554" x鈫�(0)=x鈫�0 has a unique solutionrole="math" localid="1660807989045" x鈫�(t)

(b) Show that the zero state is a stable equilibrium solution of the systemdx鈫�dt=Ax鈫� if and only if the real part of all the位i is negative.Hint: Exercise 47 and Exercise 8.1.45 are helpful.

For the values of 位1=-1and 位1=-2, sketch the trajectories for all nine initial values shown in the following figures. For each of the points, trace out both future and past of the system.

Question: Consider the interaction of two species in a habitat. We are told that the change of the populationsx鈫�(t)andy鈫�(t) can be moderated by the equation

,|dxdt=x+ydydt=2x-y|

where timeis a measured in years.

  1. What kind of intersection do we observe (symbiosis, competition, or predator-prey)?
  2. Sketch the phase portrait for the system. From the nature of the problem, we are interested only in the first quadrant.
  3. What will happen in the long term? Does the outcome depend on the initial populations? If so, how?

Solve the differential equationf''(t)+f'(t)-12f(t)=0and find solution of the differential equation.

The displacement x(t) of a certain oscillator can be modeled by the DE

d2xdt2+6dxdt+9x=0.

Find the solution x(t) for the initial value x(0) = 0,x' (0) = 1. Sketch the graph of the solution. How many times will the oscillation go through the equilibrium state x = 0in this case?
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