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
Would I do Pythagorean theorem?
kati45 [8]
Yes you do , i learned pythagorean theorem last year in pre-algebra
4 0
3 years ago
Read 2 more answers
Tell whether the given value is a solution of the inequality. (yes or no)<br><br> -2 ≤ k/3; k = -9
olga55 [171]

Answer:  No, it is not a solution

Work Shown:

-2 ≤ k/3

-2 ≤ -9/3

-2 ≤ -3

The last inequality is false because -3 should be smaller than -2 (not the other way around). Use a number line to help see this.

Since the last inequality is false, the original inequality must also be false for that particular k value. Therefore, k = -9 is not a solution.

4 0
2 years ago
480/X - 8 - 480/x = 3
Nataliya [291]

Answer:

i don't know its ans thank you

7 0
2 years ago
Read 2 more answers
Translate to an algebraic expression.
beks73 [17]
The answer is d=63/z
6 0
3 years ago
Find the exact value of tan 15 degrees
pentagon [3]
<span>tan(15) = 
sin(15) / cos(15) = 
sin(45 - 30) / cos(45 - 30) = 
[ sin(45)cos(30) - sin(30)cos(45) ] / [ cos(45)cos(30) + sin(45)sin(30)] 

Since sin(45) = cos(45) = √2/2, you can just factor that out from the top and bottom 
[ cos(30) - sin(30) ] / [ cos(30) + sin(30)] 
[ √3/2 - 1/2 ] / [ √3/2 + 1/2] 
(√3 - 1) / (√3 + 1) 
(√3 - 1)^2 / (√3+1)(√3 - 1) 
(√3 - 1)^2 / (3 - 1) 
(3 - 2√3 +1) / 2 
2 - √3 

There's also a formula for tan(a-b), but I couldn't remember it off hand.</span>
4 0
3 years ago
Other questions:
  • (−2/3)x(1/6) please answer
    8·2 answers
  • Which transformations of the graph of f(x)=-4^x result in the graph of f(x)=4^x+3?
    14·2 answers
  • If {x | x E N, x &gt; 5} then which statement is FALSE?
    12·1 answer
  • Simplify.
    10·2 answers
  • If the measure of an angle is 13 degree, find measure 0f its supplement
    8·1 answer
  • Lucas is riding his bike to the park which is 2 kilometersfrom his house. When he is one-fourth of the way tothe park, it starts
    8·1 answer
  • A college offers shuttle service from Olney Hall or Lot B to its campus quad. Both shuttles first depart their locations at ​9:1
    5·2 answers
  • Describe the overall shape of this distribution. Explain your answer in complete sentences.
    13·1 answer
  • Angles A and B are supplementary. If mA = 67, find mB.<br><br> Pleas show work!
    10·1 answer
  • Pls ill give brainelist
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!