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
AleksandrR [38]
2 years ago
7

In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y′′+4y′+3y=8te^−t+6e^−t−(9t+6)

Mathematics
1 answer:
Luden [163]2 years ago
7 0

We're given the ODE,

<em>y''</em> + 4<em>y'</em> + 3<em>y</em> = 8<em>t</em> exp(-<em>t </em>) + 6 exp(-<em>t</em> ) - (9<em>t</em> + 6)

(where I denote exp(<em>x</em>) = <em>eˣ </em>)

First determine the characteristic solution:

<em>y''</em> + 4<em>y'</em> + 3<em>y</em> = 0

has characteristic equation

<em>r</em> ² + 4<em>r</em> + 3 = (<em>r</em> + 1) (<em>r</em> + 3) = 0

with roots at <em>r</em> = -1 and <em>r</em> = -3, so the characteristic solution is

<em>y</em> = <em>C</em>₁ exp(-<em>t</em> ) + <em>C</em>₂ exp(-3<em>t</em> )

For the non-homogeneous equation, assume two ansatz solutions

<em>y</em>₁ = (<em>at</em> ² + <em>bt</em> + <em>c</em>) exp(-<em>t </em>)

and

<em>y</em>₂ = <em>at</em> + <em>b</em>

<em />

• <em>y''</em> + 4<em>y'</em> + 3<em>y</em> = 8<em>t</em> exp(-<em>t </em>) + 6 exp(-<em>t</em> ) … … … [1]

Compute the derivatives of <em>y</em>₁ :

<em>y</em>₁ = (<em>at</em> ² + <em>bt</em> + <em>c</em>) exp(-<em>t </em>)

<em>y</em>₁' = (2<em>at</em> + <em>b</em>) exp(-<em>t </em>) - (<em>at</em> ² + <em>bt</em> + <em>c</em>) exp(-<em>t </em>)

… = (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) exp(-<em>t </em>)

<em>y</em>₁'' = (-2<em>at</em> + 2<em>a</em> - <em>b</em>) exp(-<em>t </em>) - (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) exp(-<em>t </em>)

… = (<em>at</em> ² + (<em>b</em> - 4<em>a</em>) <em>t</em> + 2<em>a</em> - 2<em>b</em> + <em>c</em>) exp(-<em>t</em> )

Substitute them into the ODE [1] to get

→   [(<em>at</em> ² + (<em>b</em> - 4<em>a</em>) <em>t</em> + 2<em>a</em> - 2<em>b</em> + <em>c</em>) + 4 (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) + 3 (<em>at</em> ² + <em>bt</em> + <em>c</em>)] exp(-<em>t</em> ) = 8<em>t</em> exp(-<em>t </em>) + 6 exp(-<em>t</em> )

(<em>at</em> ² + (<em>b</em> - 4<em>a</em>) <em>t</em> + 2<em>a</em> - 2<em>b</em> + <em>c</em>) + 4 (-<em>at</em> ² + (2<em>a</em> - <em>b</em>) <em>t</em> + <em>b</em> - <em>c</em>) + 3 (<em>at</em> ² + <em>bt</em> + <em>c</em>) = 8<em>t</em> + 6

4<em>at</em> + 2<em>a</em> + 2<em>b</em> = 8<em>t</em> + 6

→   4<em>a</em> = 8   and   2<em>a</em> + 2<em>b</em> = 6

→   <em>a</em> = 2   and   <em>b</em> = 1

→   <em>y</em>₁ = (2<em>t</em> ² + <em>t </em>) exp(-<em>t </em>)

(Note that we don't find out anything about <em>c</em>, but that's okay since it would have gotten absorbed into the first characteristic solution exp(-<em>t</em> ) anyway.)

• <em>y''</em> + 4<em>y'</em> + 3<em>y</em> = -(9<em>t</em> + 6) … … … [2]

Compute the derivatives of <em>y</em>₂ :

<em>y</em>₂ = <em>at</em> + <em>b</em>

<em>y</em>₂' = <em>a</em>

<em>y</em>₂'' = 0

Substitute these into [2] :

4<em>a</em> + 3 (<em>at</em> + <em>b</em>) = -9<em>t</em> - 6

3<em>at</em> + 4<em>a</em> + 3<em>b</em> = -9<em>t</em> - 6

→   3<em>a</em> = -9   and   4<em>a</em> + 3<em>b</em> = -6

→   <em>a</em> = -3   and   <em>b</em> = 2

→   <em>y</em>₂ = -3<em>t</em> + 2

Then the general solution to the original ODE is

<em>y(t)</em> = <em>C</em>₁ exp(-<em>t</em> ) + <em>C</em>₂ exp(-3<em>t</em> ) + (2<em>t</em> ² + <em>t </em>) exp(-<em>t </em>) - 3<em>t</em> + 2

Use the initial conditions <em>y</em> (0) = 2 and <em>y'</em> (0) = 2 to solve for <em>C</em>₁ and <em>C</em>₂ :

<em>y</em> (0) = <em>C</em>₁ + <em>C</em>₂ + 2 = 2

→   <em>C</em>₁ + <em>C</em>₂ = 0 … … … [3]

<em>y'(t)</em> = -<em>C</em>₁ exp(-<em>t</em> ) - 3<em>C</em>₂ exp(-3<em>t</em> ) + (-2<em>t</em> ² + 3<em>t</em> + 1) exp(-<em>t </em>) - 3

<em>y'</em> (0) = -<em>C</em>₁ - 3<em>C</em>₂ + 1 - 3 = 2

→   <em>C</em>₁ + 3<em>C</em>₂ = -4 … … … [4]

Solve equations [3] and [4] to get <em>C</em>₁ = 2 and <em>C</em>₂ = -2. Then the particular solution to the initial value problem is

<em>y(t)</em> = -2 exp(-3<em>t</em> ) + (2<em>t</em> ² + <em>t</em> + 2) exp(-<em>t </em>) - 3<em>t</em> + 2

You might be interested in
Factor 15x + 10y =....<br> Please show how you got your answer.
Alekssandra [29.7K]

Answer:

Step-by-step explanation:

What can go into 15 and 10 both equally?

5

5 goes into 15 3 times and 5 goes into 10 twice

5(3x+2y)

HOPE THIS HELPED :3

4 0
2 years ago
Read 2 more answers
Explain how the identity property in the problem, 2/3 (1) = 1 (2/3) = 2/3, is the same with any rational number. ANSWER FAST PLE
Morgarella [4.7K]
First of all, the identity property of multiplication (which is what this is I'm assuming) is that the number 1 multiplied by any other number is that number itself. (An example would be 2 multiplied by 1, which would be two) So in this problem, this rule applies too, since 2/3 multiplied by 1 would be 2/3!

Hope this helped :)
7 0
3 years ago
Write the function as an algebraic equation
Leto [7]

Answer:

OMGGGG WANNA WRK ON THIS TOGETHER I NEED THIS PROBLEM TOOOOO

4 0
3 years ago
What are the measures of angle b e c and angle a b e in the figure shown? (30 points.)
koban [17]
Angle bec= 20 and angle abe =160
8 0
2 years ago
Read 2 more answers
Which answer best describes the shape of this distribution
Vitek1552 [10]
I think it is C.Skewed right.
5 0
3 years ago
Read 2 more answers
Other questions:
  • The vertex of parabola is (2,-2) Another point on the parabola is (5,7). find another point on the parabola.
    8·1 answer
  • The product of -2 and a number minus six is greater than -18
    5·1 answer
  • How do you recognize an exponential situation? How do you recognize an exponential equation?
    13·1 answer
  • Select the shapes that have fewer than two paírs of parallel sides
    10·1 answer
  • If two planes intersect, they intersect in exactly one ?
    6·2 answers
  • Find the slope of the line shown on the graph to the right.
    9·1 answer
  • How this is so hard plz help me on this
    10·1 answer
  • a) How does the distance traveled and time elapsed compare for Carrie and Kevin as they traveled from mile marker 225 to mile ma
    6·1 answer
  • ujeli buys trousers for rs 2000 each and price them at 2400 each . of the selling price is reduced by 20% . how much loss does s
    8·1 answer
  • One batch of gingerbread cookies uses 10 ounces of light briwn sugar. The baker uses 14 ounces of light brown sugar. How many ba
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!