1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
erastova [34]
3 years ago
5

Exercise 3.9.101: Find a particular solution to x 0 = 5x + 4y+ t, y 0 = x + 8y−t, a) using integrating factor method, b) using e

igenvector decomposition, c) using undetermined coefficients.
Mathematics
1 answer:
enot [183]3 years ago
8 0

In matrix form, the ODE is given by

\underbrace{\begin{bmatrix}x'\\y'\end{bmatrix}}_{\vec x'}=\underbrace{\begin{bmatrix}5&4\\1&8\end{bmatrix}}_A\underbrace{\begin{bmatrix}x\\y\end{bmatrix}}_{\vec x}+t\underbrace{\begin{bmatrix}1\\-1\end{bmatrix}}_{\vec f}

a. Move A\vec x to the left side and multiply both sides by the integrating factor, the matrix exponential of -A, e^{-At}:

e^{-At}\vec x'-Ae^{-At}\vec x=te^{-At}\vec f

Condense the left side as the derivative of a product:

\left(e^{-At}\vec x\right)=te^{-At}\vec f

Integrate both sides and multipy by e^{At} to solve for \vec x:

e^{-At}\vec x=\displaystyle\left(\int te^{-At}\,\mathrm dt\right)\vec f\implies\vec x=\displaystyle e^{At}\left(\int te^{-At}\,\mathrm dt\right)\vec f

Finding e^{\pm At} requires that we diagonalize A.

A has eigenvalues 4 and 9, with corresponding eigenvectors \begin{bmatrix}-4&1\end{bmatrix}^\top and \begin{bmatrix}1&1\end{bmatrix}^\top (explanation for this in part (b)), so we have

A=\begin{bmatrix}-4&1\\1&1\end{bmatrix}\begin{bmatrix}4&0\\0&9\end{bmatrix}\begin{bmatrix}-4&1\\1&1\end{bmatrix}^{-1}

\implies A^n=\begin{bmatrix}-4&1\\1&1\end{bmatrix}\begin{bmatrix}4^n&0\\0&9^n\end{bmatrix}\begin{bmatrix}-4&1\\1&1\end{bmatrix}^{-1}

\implies A^n=\dfrac15\begin{bmatrix}4^{n+1}+9^n&4\cdot9^n-4^{n+1}\\9^n-4^n&4^n+4\cdot9^n\end{bmatrix}

\implies e^{\pm At}=\dfrac15\begin{bmatrix}4e^{\pm4t}+e^{\pm9t}&4e^{\pm9t}-4e^{\pm4t}\\e^{\pm9t}-e^{\pm4t}&e^{\pm4t}+4e^{\pm9t}\end{bmatrix}

\implies\vec x=\dfrac15e^{At}\begin{bmatrix}C_1\\C_2\end{bmatrix}-\dfrac1{216}\begin{bmatrix}72t+20\\-36t-7\end{bmatrix}

b. Find the eigenvalues of A:

\det(A-\lambda I_2)=\begin{vmatrix}5-\lambda&4\\1&8-\lambda\end{vmatrix}=\lambda^2-13\lambda+36=0

\implies(\lambda-4)(\lambda-9)=0\implies\lambda_1=4,\lambda_2=9

Let \vec\eta=\begin{bmatrix}\eta_1&\eta_2\end{bmatrix}^\top and \vec\theta=\begin{bmatrix}\theta_1&\theta_2\end{bmatrix}^\top be the corresponding eigenvectors.

For \lambda_1=4, we have

\begin{bmatrix}1&4\\1&4\end{bmatrix}\begin{bmatrix}\eta_1\\\eta_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}

which means we can pick \eta_1=-4 and \eta_2=1.

For \lambda_2=9, we have

\begin{bmatrix}-4&4\\1&-1\end{bmatrix}\begin{bmatrix}\theta_1\\\theta_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}

so we pick \theta_1=\theta_2=1.

Then the characteristic solution to the system is

\vec x_c=C_1e^{\lambda_1t}\vec\eta+C_2e^{\lambda_2t}\vec\theta

\vec x_c=C_1e^{4t}\begin{bmatrix}-4\\1\end{bmatrix}+C_2e^{9t}\begin{bmatrix}1\\1\end{bmatrix}

c. Now we find the particular solution with undetermined coefficients.

The nonhomogeneous part of the ODE is a linear function, so we can start with assuming a particular solution of the form

\vec x_p=\vec at+\vec b\implies\vec x_p'=\vec a

Substituting these into the system gives

\begin{bmatrix}a_1\\a_2\end{bmatrix}=\begin{bmatrix}5&4\\1&8\end{bmatrix}\left(\begin{bmatrix}a_1\\a_2\end{bmatrix}t+\begin{bmatrix}b_1\\b_2\end{bmatrix}\right)+\begin{bmatrix}1\\-1\end{bmatrix}t

\begin{bmatrix}a_1\\a_2\end{bmatrix}=\begin{bmatrix}5&4\\1&8\end{bmatrix}\begin{bmatrix}a_1t+b_1\\a_2t+b_2\end{bmatrix}+\begin{bmatrix}t\\-t\end{bmatrix}

\begin{bmatrix}a_1\\a_2\end{bmatrix}=\begin{bmatrix}(5a_1+4a_2+1)t+(5b_1+4b_2)\\(a_1+8a_2-1)t+(b_1+8b_2)\end{bmatrix}

\implies\begin{cases}5a_1+4a_2=-1\\5b_1+4b_2=a_1\\a_1+8a_2=1\\b_1+8b_2=a_2\end{cases}\implies a_1=-\dfrac13,a_2=\dfrac16,b_1=-\dfrac5{54},b_2=\dfrac7{216}

Put everything together to get a solution

\vec x=\vec x_c+\vec x_p

that should match the solution in part (a).

You might be interested in
You inherit $1000, but spends $8 each day at lunch. How many days until you have $768
schepotkina [342]

Answer:

29 days

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
How did the gulf of Tonkin resolutions affect US involvement in the Vietnam war?
klio [65]

Answer:

I don't know how this is math but...

The Gulf of Tonkin Resolution authorized President Lyndon Johnson to “take all necessary measures to repel any armed attack against the forces of the United States and to prevent further aggression” by the communist government of North Vietnam. Hope I helped! ☺

7 0
2 years ago
write the equation of a line that includes the point (1,5) and has a slope of 3 in slope intercept form.
lina2011 [118]
Y=3x+? (1,5)
5=3(1)+?
5=3+?

y=3x+2

7 0
3 years ago
Select the ordered pair that is a solution to the function. (A point on the line.)?
soldier1979 [14.2K]

Answer:

-7

Step-by-step explanation:

first you plug in f(3) in the x so -2(3)= -6

then -6 - 1 = -7

6 0
2 years ago
Which of these is not a Pythagorean Triple?
Greeley [361]

Answer:

C because the rest are pythagoreaum theroems 1 squared plus 2 squared does not equal 3 squared.

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Other questions:
  • A rope is 440 cm long is it in kilometers
    11·2 answers
  • 6x +2 =2(x+4)-2 helpp meee
    13·2 answers
  • What are the domain and range of the function f(x)=3x-1 /2
    12·1 answer
  • HELP ASAP GIVING BRSINLIST
    9·1 answer
  • 24=3r divided by 4. what is the value of r?
    14·2 answers
  • What is the solution of 5x-6=44
    6·2 answers
  • Determine how many solutions the equation has. If one solution, state the value of x.
    11·1 answer
  • What is the value of X?
    8·2 answers
  • 80% of the students in Lea’s class earned an A or a B on the weekly math test. If 20 students got an A or a B, how many students
    9·2 answers
  • Don't understand letter K can u write the equation of the described function in standard form
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!