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]
3 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]3 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
Help please, I don’t understand this question.
Hatshy [7]

Answer:

all you have to do is find x and then what ever you get for x plug it back in where the y is to find the y value

4 0
3 years ago
You are considering purchasing a new home.
dexar [7]
Answer the answer is B) 990
3 0
3 years ago
Rewrite <br><img src="https://tex.z-dn.net/?f=%20%7Bcos%7D%5E%7B4%7D%20x" id="TexFormula1" title=" {cos}^{4} x" alt=" {cos}^{4}
erastovalidia [21]

All you need is the double angle identity:

\cos^2x=\dfrac{1+\cos2x}2

So we have

\cos^4x=(\cos^2x)^2=\left(\dfrac{1+\cos2x}2\right)^2=\dfrac{1+2\cos2x+\cos^22x}4

Apply the identity again to the squared term:

\cos^4x=\dfrac{1+2\cos2x+\frac{1+\cos4x}2}4=\dfrac{3+4\cos2x+\cos4x}8

6 0
3 years ago
R^4/9 translated verbal expression
Verizon [17]

Answer:

The quotient of 4 and 9 raised to the power of r.

Step-by-step explanation:

3 0
3 years ago
Using data from a study, we find a significant difference in the proportion of fruit flies surviving after 13 days between those
Alchen [17]

Answer:

uns 20 a 25%

Step-by-step explanation:

5 0
3 years ago
Other questions:
  • ( 5x^4- 3x^2 - 4x+6)/(x-7)
    15·1 answer
  • Please help. A spinner had 7 identical sections. Two sections are blue, 1 us red, and 4 of the sections are green. Suppose the p
    8·1 answer
  • Please help me out with this problem!
    6·1 answer
  • Convert .04 to percent
    8·2 answers
  • I NEED HELP ASAP!!! Suppose a study shows that 1 out of 287 computers are hacked. If a company has 8,564 computers, how many are
    14·2 answers
  • Help needed: Solve for x. Round to the nearest tenth, if necessary.
    15·1 answer
  • Please help me with this question
    15·1 answer
  • I need help with this:)​
    9·2 answers
  • Trains leave Bristol to Cardiff every 15 minutes to London every 21 minutes A train to Cardiff and a train to London both leave
    14·1 answer
  • Ayva earned a 3% commission on a sale of
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!