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
shepuryov [24]
3 years ago
5

Solve the given initial-value problem. x' = 1 2 0 1 − 1 2 x, x(0) = 2 7

Mathematics
1 answer:
Ilia_Sergeevich [38]3 years ago
5 0
I'll go out on a limb and guess the system is

\mathbf x'=\begin{bmatrix}\frac12&0\\1&-\frac12\end{bmatrix}\mathbf x

with initial condition \mathbf x(0)=\begin{bmatrix}2&7\end{bmatrix}^\top. The coefficient matrix has eigenvalues \lambda such that

\begin{vmatrix}\frac12-\lambda&0\\1&-\frac12-\lambda\end{vmatrix}=\lambda^2-\dfrac14=0\implies\lambda=\pm\dfrac12

The corresponding eigenvectors \eta are such that

\lambda=\dfrac12\implies\begin{bmatrix}\frac12-\frac12&0\\1&-\frac12-\frac12\end{bmatrix}\eta=\begin{bmatrix}0&0\\1&-1\end{bmatrix}\eta=\begin{bmatrix}0\\0\end{bmatrix}
\implies\eta=\begin{bmatrix}1\\1\end{bmatrix}

\lambda=-\dfrac12\implies\begin{bmatrix}\frac12+\frac12&0\\1&-\frac12+\frac12\end{bmatrix}\eta=\begin{bmatrix}1&0\\1&0\end{bmatrix}\eta=\begin{bmatrix}0\\0\end{bmatrix}
\implies\eta=\begin{bmatrix}0\\1\end{bmatrix}

So the characteristic solution to the ODE system is

\mathbf x(t)=C_1\begin{bmatrix}1\\1\end{bmatrix}e^{t/2}+C_2\begin{bmatrix}0\\1\end{bmatrix}e^{-t/2}

When t=0, we have

\begin{bmatrix}2\\7\end{bmatrix}=C_1\begin{bmatrix}1\\1\end{bmatrix}+C_2\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}C_1\\C_1+C_2\end{bmatrix}

from which it follows that C_1=2 and C_2=5, making the particular solution to the IVP

\mathbf x(t)=2\begin{bmatrix}1\\1\end{bmatrix}e^{t/2}+5\begin{bmatrix}0\\1\end{bmatrix}e^{-t/2}

\mathbf x(t)=\begin{bmatrix}2e^{t/2}\\2e^{t/2}+5e^{-t/2}\end{bmatrix}
You might be interested in
Simon, 35, works for a company which offers a 4% match to his 401(k) plan. He previously invested $25,000 into an IRA. What woul
Paha777 [63]

Answer: answer

Step-by-step explanation:

4 0
3 years ago
Solve for x. x2 + 7x + 10 = 0
algol13
The answer is negative 80 or like this -80
6 0
3 years ago
Identify the law illustrated by: 4(3x-7) is equivalent to 12x-28.
11Alexandr11 [23.1K]

Answer:

Distributive law of multiplication

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
When Mr. Key made a chocolate milkshake for himself, he used 3/4 cup
icang [17]

Answer:

3 cups of milk

Step-by-step explanation:

4/4 is 1 cup.

8 0
3 years ago
A grocery store is running a special on frozen dinners. If a customer buys a family-size dinner
qaws [65]

Answer:

2x + 8

Step-by-step explanation:

This is the pattern:

y is 2 times x plus 8

Write this relationship as an equation.

y=2x+8

7 0
2 years ago
Other questions:
  • Can someone help I too lazy
    7·2 answers
  • Write the rule for the table below help me pleaseeeeeeeeee
    15·1 answer
  • In a research sample of high school students, the variable “number of Instagram followers” formed normal distribution with a mea
    5·1 answer
  • What is the answer of 2+4+4567
    6·2 answers
  • Add the two mixed numbers 1 1/3 + 3 2/3 = ?
    13·1 answer
  • What is the greatest number which can be made using each of the digits 5,3,1,4,7?
    11·1 answer
  • PLEASE HELP ME !!!!!!!!!!! Will mark Brianliest correct answer !!!!!!!!!!!!!!!!!!!!!!!!
    15·1 answer
  • Please help due today!
    7·1 answer
  • a driver is below the surface of the water at - 15m. he dives down by 6m, then rises 4m where is he now​
    9·1 answer
  • A. The ratio of the number of boys
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!