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
zheka24 [161]
3 years ago
14

Y"-4y'+4y=0 y1=e^2x find the general solution

Mathematics
1 answer:
Karo-lina-s [1.5K]3 years ago
4 0
Given that y_1=e^{2x} is a known solution to the ODE, we can reduce the order of the ODE to find a second solution of the form

y_2=y_1v=e^{2x}v
\implies {y_2}'=2e^{2x}v+e^{2x}v'
\implies {y_2}''=4e^{2x}v+4e^{2x}v'+e^{2x}v''

Substituting into the ODE, we get

(e^{2x}v''+4e^{2x}v'+4e^{2x}v)-4(e^{2x}v'+2e^{2x}v)+4e^{2x}v=e^{2x}
e^{2x}v''=e^{2x}
v''=1
\implies v'=C_1
\implies v=C_1x+C_2
\implies y_2=C_1xe^{2x}+C_2e^{2x}

We already know about e^{2x} as a solution to the ODE, which means y_2=xe^{2x}.

That covers the characteristic solution. To find the particular solution to the nonhomogeneous ODE, suppose there is a solution of the form

y_p=ax^2e^{2x}
{y_p}'=2ax(x+1)e^{2x}
{y_p}''=2a(2x^2+4x+1)e^{2x}

Substituting into the ODE yields

{y_p}''-4{y_p}'+4{y_p}=2ae^{2x}=e^{2x}\implies a=\dfrac12

so that the general solution is

y=C_1y_1+C_2y_2+y_p
y=C_1e^{2x}+C_2xe^{2x}+\dfrac12x^2e^{2x}
You might be interested in
How do I find the displacement? Help frl
liubo4ka [24]

Answer:

  • distance traveled: 30 m
  • displacement: 21.4 m

Step-by-step explanation:

You want the distance traveled and the displacement after walking 17 m south and 13 m east.

<h3>Distance</h3>

The distance traveled is the sum of the lengths of each leg of the trip:

  17 m + 13 m = 30 m

You have traveled a distance of 30 m.

<h3>Displacement</h3>

The displacement is the distance from your final position to your starting position. If you draw a diagram of the journey, you see the displacement is the hypotenuse of a right triangle with legs 17 m and 13 m. The Pythagorean theorem can help you find this length:

  h = √(a² +b²)

  h = √(17² +13²) = √(289 +169) = √458 ≈ 21.401

At the end of your walking, you are 21.4 m from where you started.

5 0
2 years ago
Which value of x makes the equation 1.25(6x - 20) = 50 true?
bixtya [17]
D. X = 10

6x-20= 40

6x=40+20

6x=60

X=10
4 0
3 years ago
HELP ASAP. Determine whether triangles are similar by Side by Side (SSS), Side Angle Side (SAS), or Angle Angle (AA) similarity.
Nonamiya [84]
The answer and process are in the picture.

4 0
2 years ago
In which step did fioana make an error
sladkih [1.3K]

What are the steps? Is there a photo?


5 0
3 years ago
Read 2 more answers
Your mom purchased an antique lamp for $245. The value
larisa86 [58]

Answer:

y=245 * 1.035^t   $487.50

Step-by-step explanation:

y=245*1.035^20

y=245*1.9898

y=487.49827

6 0
3 years ago
Other questions:
  • The speaker is the ____________ of the message.
    14·1 answer
  • Farmer Ed has 8,000 meters of fencing &amp; wants to enclose a rectangular plot that borders a river. If farmer Ed does not fenc
    6·1 answer
  • A random sample of 4000 U.S. citizens yielded 2150 who are in favor of outlawing cheesing. Find the point estimate for estimatin
    7·1 answer
  • The expression (x-6)^2
    6·2 answers
  • The price of all rail season tickets to London increased by 4%. The price of a rail season ticket from Cambridge to London incre
    13·1 answer
  • I am not sure how to do number 10?
    7·1 answer
  • Please help with the image below
    8·2 answers
  • Health of the intersection
    5·1 answer
  • Você precisa colocar suas renas, Raposa, Trovão, Cometa e Relâmpago, em uma fila única para puxar o seu trenó. No entanto, Comet
    11·1 answer
  • Fill in the blank (no word bank)
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!