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
Vikentia [17]
3 years ago
15

Consider the following differential equation to be solved by undetermined coefficients. y(4) − 2y''' + y'' = ex + 1 Write the gi

ven differential equation in the form L(y) = 0, where L is a linear operator with constant coefficients. If possible, factor L. (Use D for the differential operator.) Incorrect: Your answer is incorrect. y = ex + 1 Find a linear differential operator that annihilates the function g(x) = ex + 1. (Use D for the differential operator.) Solve the given differential equation by undetermined coefficients. y(x) =
Mathematics
1 answer:
kompoz [17]3 years ago
3 0

Answer:

The general solution is

y= (C_{1}+C_{1}x) e^0x+(C_{3}+C_{4}x) e^x +\frac{1}{2} (e^x(x^2-2x+2)-e^x(2(x-1)+e^x(2))

     + \frac{x^2}{2}

Step-by-step explanation:

Step :1:-

Given differential equation  y(4) − 2y''' + y'' = e^x + 1

The differential operator form of the given differential equation

(D^4 -2D^3+D^2)y = e^x+1

comparing f(D)y = e^ x+1

The auxiliary equation (A.E) f(m) = 0

                         m^4 -2m^3+m^2 = 0

                         m^2(m^2 -2m+1) = 0

(m^2 -2m+1) this is the expansion of (a-b)^2

                        m^2 =0 and (m-1)^2 =0

The roots are m=0,0 and m =1,1

complementary function is y_{c} = (C_{1}+C_{1}x) e^0x+(C_{3}+C_{4}x) e^x

<u>Step 2</u>:-

The particular equation is    \frac{1}{f(D)} Q

P.I = \frac{1}{D^2(D-1)^2} e^x+1

P.I = \frac{1}{D^2(D-1)^2} e^x+\frac{1}{D^2(D-1)^2}e^{0x}

P.I = I_{1} +I_{2}

\frac{1}{D^2} (\frac{x^2}{2!} )e^x + \frac{1}{D^{2} } e^{0x}

\frac{1}{D} means integration

\frac{1}{D^2} (\frac{x^2}{2!} )e^x = \frac{1}{2D} \int\limits {x^2e^x} \, dx

applying in integration u v formula

\int\limits {uv} \, dx = u\int\limits {v} \, dx - \int\limits ({u^{l}\int\limits{v} \, dx  } )\, dx

I_{1} = \frac{1}{D^2(D-1)^2} e^x

\frac{1}{2D} (e^x(x^2)-e^x(2x)+e^x(2))

\frac{1}{2} (e^x(x^2-2x+2)-e^x(2(x-1)+e^x(2))

I_{2}= \frac{1}{D^2(D-1)^2}e^{0x}

\frac{1}{D} \int\limits {1} \, dx= \frac{1}{D} x

again integration  \frac{1}{D} x = \frac{x^2}{2!}

The general solution is y = y_{C} +y_{P}

         y= (C_{1}+C_{1}x) e^0x+(C_{3}+C_{4}x) e^x +\frac{1}{2} (e^x(x^2-2x+2)-e^x(2(x-1)+e^x(2))

      + \frac{x^2}{2!}

You might be interested in
Mr. Mole left his burrow and started digging his way down at a constant rate.
Iteru [2.4K]

Constant rate means it is linear so we don't have to worry about fluctuation.

change in y/ change in x=-18-(-25.2)/5-8=2.4 meters per second

6 0
3 years ago
Read 2 more answers
X2y − 7xy + xyz + x Classify the following expression by degree and term
Dmitriy789 [7]
Polynomial degree 3 
Term xyz
Coefficient 1
7 0
2 years ago
What is the quotient of 2 3/4 and 5 1/2?
Juli2301 [7.4K]

Answer:

  1/2

Step-by-step explanation:

This can be done several ways. Perhaps the easiest is to use the decimal equivalents:

  (2 3/4)/(5 1/2) = 2.75/5.5 = 0.5

__

If you want to use the numbers given, the usual procedure is to convert them to improper fractions and do the division that way:

  2 3/4 = (4·2 +3)/4 = 11/4

  5 1/2 = (2·5 +1)/2 = 11/2

Now, the problem can be written as ...

  (2 3/4) / (5 1/2) = (11/4) / (11/2)

This sort of division problem can be solved two ways:

  <u>invert and multiply</u> (the denominator is inverted)

  = (11/4) × (2/11) = (11·2)/(11·4) = 2/4 = 1/2

  <u>use a common denominator</u> (for the two fractions)

  = (11/4) / (11/2) = (11/4) / (22/4) = 11/22 = 1/2

The quotient is 1/2.

_____

<em>Additional comments</em>

It is helpful in many cases to just use decimal equivalents for the calculation. To do that, you need to be familiar with the equivalents of commonly used fractions. I find it is usually sufficient to know the unit fractions in each case. Then you can multiply or add to find the others.

1/9 = 0.1...(1-digit repeat), 1/8 = 0.125, 1/7 = 0.142857...(6-digit repeat), 1/6 = 0.16...(1-digit repeat), 1/5 = 0.2, 1/4 = 0.25, 1/3 = 0.3...(1-digit repeat), 1/2 = 0.5

__

I learned the "invert and multiply" method for dividing fractions when I was in school. The Common Core math apparently also teaches the method of matching the denominators of a compound fraction. Then the result is the ratio of numerators. (A variation not taught is that you can match the numerators and use the inverse of the ratio of denominators. Both of these methods can be validated using the "invert and multiply" method. (11/4)/(11/2) = 2/4)

"Invert and multiply" is another way of saying that division is the same as multiplication by the reciprocal. This applies everywhere, not just in fraction problems.

8 0
3 years ago
A=12/35<br> B=35/37<br> C=35/12<br> D=12/37
Scrat [10]

Answer: D= 12/37

Step-by-step explanation:

To find sin of angle A, you take the opposite angle over the hypotenuse. 12 is opposite of A and 37is the hypotenuse.

3 0
3 years ago
Read 2 more answers
All help is appreciated, thanks everyone in advance *Slopes :)
Katena32 [7]
This one is a bit more complicated but i believe the answer is -1/4

4 0
2 years ago
Other questions:
  • In one month, Jillian made 36 local phone calls and 20 long distance calls. What was her ratio of local calls to long distance c
    11·2 answers
  • Round 78.932 to the ones,tenths,and hundredths place.​
    6·1 answer
  • A single die is rolled 100 times. list the probabilities below from least to greatest. p(odd), p(&gt;4), p(prime), p(8)
    11·1 answer
  • Como puedo resolver este binomio <br> (10-m)3
    11·1 answer
  • The tip jar at a sandwich shop has been found to have a dollar value 0.65ℎ +1.25, where ℎ is the number of hours since the shop
    10·1 answer
  • (the stake is 14ft)
    8·1 answer
  • A grain trader buys the following amounts from three suppliers: 3,200 pounds, 7,200 pounds and 600 pounds. What is the total wei
    15·2 answers
  • A or B? I need the answer soon as possible.
    12·1 answer
  • What is the y-intercept of the line graphed on the grid?
    13·2 answers
  • Serena hits a tennis ball downward from the top of the net at which the angle of
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!