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
kupik [55]
2 years ago
9

Apply the method of undetermined coefficients to find a particular solution to the following system.wing system.

Mathematics
1 answer:
jarptica [38.1K]2 years ago
7 0
  • y''-y'+y=\sin x

The corresponding homogeneous ODE has characteristic equation r^2-r+1=0 with roots at r=\dfrac{1\pm\sqrt3}2, thus admitting the characteristic solution

y_c=C_1e^x\cos\dfrac{\sqrt3}2x+C_2e^x\sin\dfrac{\sqrt3}2x

For the particular solution, assume one of the form

y_p=a\sin x+b\cos x

{y_p}'=a\cos x-b\sin x

{y_p}''=-a\sin x-b\cos x

Substituting into the ODE gives

(-a\sin x-b\cos x)-(a\cos x-b\sin x)+(a\sin x+b\cos x)=\sin x

-b\cos x+a\sin x=\sin x

\implies a=1,b=0

Then the general solution to this ODE is

\boxed{y(x)=C_1e^x\cos\dfrac{\sqrt3}2x+C_2e^x\sin\dfrac{\sqrt3}2x+\sin x}

  • y''-3y'+2y=e^x\sin x

\implies r^2-3r+2=(r-1)(r-2)=0\implies r=1,r=2

\implies y_c=C_1e^x+C_2e^{2x}

Assume a solution of the form

y_p=e^x(a\sin x+b\cos x)

{y_p}'=e^x((a+b)\cos x+(a-b)\sin x)

{y_p}''=2e^x(a\cos x-b\sin x)

Substituting into the ODE gives

2e^x(a\cos x-b\sin x)-3e^x((a+b)\cos x+(a-b)\sin x)+2e^x(a\sin x+b\cos x)=e^x\sin x

-e^x((a+b)\cos x+(a-b)\sin x)=e^x\sin x

\implies\begin{cases}-a-b=0\\-a+b=1\end{cases}\implies a=-\dfrac12,b=\dfrac12

so the solution is

\boxed{y(x)=C_1e^x+C_2e^{2x}-\dfrac{e^x}2(\sin x-\cos x)}

  • y''+y=x\cos(2x)

r^2+1=0\implies r=\pm i

\implies y_c=C_1\cos x+C_2\sin x

Assume a solution of the form

y_p=(ax+b)\cos(2x)+(cx+d)\sin(2x)

{y_p}''=-4(ax+b-c)\cos(2x)-4(cx+a+d)\sin(2x)

Substituting into the ODE gives

(-4(ax+b-c)\cos(2x)-4(cx+a+d)\sin(2x))+((ax+b)\cos(2x)+(cx+d)\sin(2x))=x\cos(2x)

-(3ax+3b-4c)\cos(2x)-(3cx+3d+4a)\sin(2x)=x\cos(2x)

\implies\begin{cases}-3a=1\\-3b+4c=0\\-3c=0\\-4a-3d=0\end{cases}\implies a=-\dfrac13,b=c=0,d=\dfrac49

so the solution is

\boxed{y(x)=C_1\cos x+C_2\sin x-\dfrac13x\cos(2x)+\dfrac49\sin(2x)}

You might be interested in
Sam invest $8333 into a savings account with a compound interest of 2.7% for 10 years what is Sams balance after 10 years
devlian [24]

Answer: 10,582.91

Step-by-step explanation:  2.7%*10=27% 8,333*127%= $10,582.91

6 0
2 years ago
Read 2 more answers
Jim and Jackie are married with three children at home and a mortgage. Jim’s net pay per year is $67,000 and Jackie does not hav
rodikova [14]

The correct option is that they should work on their plan for managing income.

<h3>How to illustrate the information?</h3>

Given that Jim and Jackie are married with three children and have a home at mortgage.

The net pay per year of Jim is $67,000 and Jackie do not earn at all. The mortgage expenses and monthly expenses are

$(2,800+2,700) = $5,500, not much as compared to the total income.

Also, Jim contributes 15% of his income , i.e., $10050 into the retirement fund and also they have savings of $5,000.

In addition, there is a $500,000 life insurance policy on Jim and a $100,000 policy on Jackie.

Therefore, looking at all of these, it can be concluded that the net income of Jim per year is not bad at all to manage the expenses, they just have to work on it little more to run the family smoothly.

Thus, the correct option is that they should work on their plan for managing income.

Learn more about income on:

brainly.com/question/25745683

#SPJ1

Complete question:

Jim and Jackie are married with three children at home and a mortgage. Jim’s net pay per year is $67,000 and Jackie does not have income. Their mortgage payment of $2,800 includes insurance on their home. They have additional monthly expenses of $2,700. Jim contributes 15% of his earnings to a retirement fund and they have $5,000 in savings. There is a $500,000 life insurance policy on Jim and a $100,000 policy on Jackie. As their financial advisor, what part of Jim and Jackie’s financial plan would you encourage them to work on? a. They should work on their plan for managing income. b. They should work on their plan for managing their liquidity. c. They should work on their plan for protecting their assets. d. They should work on their plan for protecting their income.

3 0
1 year ago
Plz help!!
inysia [295]
<span>1. 5564÷91
I know that 9 * 6 = 56
5564 rounds to 5600
91 rounds to 9
Since 56/9 = 6, then 5600/90 is the same as 560/9 = 60
The estimate is 60

2. </span><span>5391÷25
5391 sounds to 5400
25 is 1/4 of 100.
That means when you divide by 25, you can divide by 100 and multiply by 4.
5400/100 = 54
54 * 4 = 216
Estimate: 216

3. </span><span>explain how to estimate 498÷12
48/12 = 4
498 is little more than 480, so 498/12 is little more than 40

4. </span><span>which is the closest estimate for 2130÷ 33
A.7 B.17 C.70 D.700

2130/33
Round off the numerator and denominator to
2100/30
Reduce the fraction
210/3
Since I know that 21/3 = 7, then 210/3 = 70
Estimate: 70
</span>
8 0
3 years ago
If trianglePTS ~ trianglePQR, find the value of x.
tresset_1 [31]

Answer:

bei2uer22vv

Step-by-step explanation:

8 0
2 years ago
Use function notation to write a recursive formula to represent the sequence: 4, 8, 12, …
kogti [31]

Answer:

  • f(1) = 4
  • f(n) = f(n - 1) + 4

Step-by-step explanation:

Given the sequence:

  • 4, 8, 12, …

The first term is 4 and common difference is 4

<u>Recursive formula:</u>

  • f(1) = 4
  • f(n) = f(n - 1) + 4

7 0
2 years ago
Other questions:
  • 2,000 water bottles are being labeled and boxed. Each box holds 6 water bottles. How many bottles will be left over?
    6·1 answer
  • How do I solve 5(4x -5) = 155
    11·1 answer
  • 14 is away from the other data. What is 14 called?
    6·1 answer
  • F(x)=2-3x
    13·1 answer
  • Help me :((
    5·2 answers
  • Find all the numbers such that the number added to itself is less than the number subtracted from 6
    11·1 answer
  • What is the mean absolute deviation?<br> 92, 83, 88, 94, 91, 85, 89, 90
    11·1 answer
  • Which polynomial correctly combines the like terms and puts the given polynomial in standard form? –5x3y3 8x4y2 – xy5 – 2x2y4 8x
    9·1 answer
  • A. y= 1/2x-14<br> B. y=1/2x-5<br> C. y=2x-14<br> D. y=2x-5
    6·1 answer
  • To Enter the value of n that makes the equation: 4^5.4^n=4^8 true
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!