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
Leto [7]
1 year ago
9

Use the method of undetermined coefficients to find the general solution to the de y′′−3y′ 2y=ex e2x e−x

Mathematics
1 answer:
djverab [1.8K]1 year ago
6 0

I'll assume the ODE is

y'' - 3y' + 2y = e^x + e^{2x} + e^{-x}

Solve the homogeneous ODE,

y'' - 3y' + 2y = 0

The characteristic equation

r^2 - 3r + 2 = (r - 1) (r - 2) = 0

has roots at r=1 and r=2. Then the characteristic solution is

y = C_1 e^x + C_2 e^{2x}

For nonhomogeneous ODE (1),

y'' - 3y' + 2y = e^x

consider the ansatz particular solution

y = axe^x \implies y' = a(x+1) e^x \implies y'' = a(x+2) e^x

Substituting this into (1) gives

a(x+2) e^x - 3 a (x+1) e^x + 2ax e^x = e^x \implies a = -1

For the nonhomogeneous ODE (2),

y'' - 3y' + 2y = e^{2x}

take the ansatz

y = bxe^{2x} \implies y' = b(2x+1) e^{2x} \implies y'' = b(4x+4) e^{2x}

Substitute (2) into the ODE to get

b(4x+4) e^{2x} - 3b(2x+1)e^{2x} + 2bxe^{2x} = e^{2x} \implies b=1

Lastly, for the nonhomogeneous ODE (3)

y'' - 3y' + 2y = e^{-x}

take the ansatz

y = ce^{-x} \implies y' = -ce^{-x} \implies y'' = ce^{-x}

and solve for c.

ce^{-x} + 3ce^{-x} + 2ce^{-x} = e^{-x} \implies c = \dfrac16

Then the general solution to the ODE is

\boxed{y = C_1 e^x + C_2 e^{2x} - xe^x + xe^{2x} + \dfrac16 e^{-x}}

You might be interested in
120 to 52 what % decrease
noname [10]
The formula is percent= part/whole so it would be 52/120 which is .43333333333<span> or 43%</span>
3 0
3 years ago
Use the following diagram to answer question 4 and 5
baherus [9]

Answer:

3x-13+2x+4=116( the sum of two interior angle is epual to the sum of exterior angle )

5x =116-9

5x=107

x=21.4 degree

then

angle a= 3*21.4-13

a=51.2degree

angle b=2*21.4+2

b =44.8degree

Step-by-step explanation:

plz make me brainliest

3 0
2 years ago
Antonio ate 9/10 of a pizza on Monday. On Tuesday, he ate 3/10 of a pizza. How much pizza did Antonio eat on Monday and Tuesday?
AveGali [126]

He ate 1 1/5  (1.2) of pizza

6 0
3 years ago
a fish tank is in the shape of a cube the. volume of the fish tank is 64 cubic feet. What is the area of one face of the fish ta
babymother [125]
16 square feet. The cube root of 64 is 4, which is equal to the length of one side. The area of one side is the length times the width, and since this is a cube, the length equals the width, meaning that the side is 4 times 4 which is 16
3 0
3 years ago
Read 2 more answers
Write an expression to show the costs of 2 sandwiches 2 sandwiches 2 drinks and a salad .how many terms does your expression hav
Nataly [62]
I would say two
Have a good day
7 0
3 years ago
Other questions:
  • I just signed up and when I put in a question in the Question Box it blocks me from seeing the answering and I wants me to sign
    14·1 answer
  • What are the possible solutions to 33⋅3333⋅33, if 11⋅11=411⋅11=4 and 22⋅22=1622⋅22=16?
    13·1 answer
  • If the new athletic fee is $22 and the old fee is $20, how much is the percent increase in fees?
    8·2 answers
  • HELP QUICK! PLEASE, 10+5 POINTS AND BRAINLIST!
    10·1 answer
  • What is the solution to the system of equations y=-3×-2 5×+2y=15
    7·2 answers
  • Let a, b, and c be constants, and let x be a variable. Which of the following is
    11·1 answer
  • Natasha is a bank teller. She received a 5% raise last year and a subsequent merit raise of 3% this year. What is the accumulati
    11·1 answer
  • I need people to talk too ughhhh comment plzz
    9·1 answer
  • 2. Tell whether the ordered pair (-1,-2) is the solution of the system. x + y =
    5·1 answer
  • Happy (Valentine) day!
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!