Some advantages of consolidating your debt into a single loan are ________

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

Luden [163]

We're given the ODE,

y'' + 4y' + 3y = 8t exp(-t ) + 6 exp(-t ) - (9t + 6)

(where I denote exp(x) = eˣ )

First determine the characteristic solution:

y'' + 4y' + 3y = 0

has characteristic equation

r ² + 4r + 3 = (r + 1) (r + 3) = 0

with roots at r = -1 and r = -3, so the characteristic solution is

y = C₁ exp(-t ) + C₂ exp(-3t )

For the non-homogeneous equation, assume two ansatz solutions

y₁ = (at ² + bt + c) exp(-t )

and

y₂ = at + b

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

Compute the derivatives of y₁ :

y₁ = (at ² + bt + c) exp(-t )

y₁' = (2at + b) exp(-t ) - (at ² + bt + c) exp(-t )

… = (-at ² + (2a - b) t + b - c) exp(-t )

y₁'' = (-2at + 2a - b) exp(-t ) - (-at ² + (2a - b) t + b - c) exp(-t )

… = (at ² + (b - 4a) t + 2a - 2b + c) exp(-t )

Substitute them into the ODE [1] to get

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

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

4at + 2a + 2b = 8t + 6

→ 4a = 8 and 2a + 2b = 6

→ a = 2 and b = 1

→ y₁ = (2t ² + t ) exp(-t )

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

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

Compute the derivatives of y₂ :

y₂ = at + b

y₂' = a

y₂'' = 0

Substitute these into [2] :

4a + 3 (at + b) = -9t - 6

3at + 4a + 3b = -9t - 6

→ 3a = -9 and 4a + 3b = -6

→ a = -3 and b = 2

→ y₂ = -3t + 2

Then the general solution to the original ODE is

y(t) = C₁ exp(-t ) + C₂ exp(-3t ) + (2t ² + t ) exp(-t ) - 3t + 2

Use the initial conditions y (0) = 2 and y' (0) = 2 to solve for C₁ and C₂ :

y (0) = C₁ + C₂ + 2 = 2

→ C₁ + C₂ = 0 … … … [3]

y'(t) = -C₁ exp(-t ) - 3C₂ exp(-3t ) + (-2t ² + 3t + 1) exp(-t ) - 3

y' (0) = -C₁ - 3C₂ + 1 - 3 = 2

→ C₁ + 3C₂ = -4 … … … [4]

Solve equations [3] and [4] to get C₁ = 2 and C₂ = -2. Then the particular solution to the initial value problem is

y(t) = -2 exp(-3t ) + (2t ² + t + 2) exp(-t ) - 3t + 2

7 0

2 years ago

Can somebody please help me out please my grads are bad

Schach [20]

Step-by-step explanation:

Starting from the upper left in the clockwise direction:

congruent: no

similar: no

upper right side:

congruent: yes

similar: yes

lower right side:

congruent: no

similar: yes

lower left side:

congruent: yes

similar: yes

7 0

2 years ago

you are on a 750 mile trip by car . ur car used seven gallons of gas to travel the first 210 miles of the trip. how much more ga

stepan [7]

Answer:

A total of 18 gallons of more gas

Step-by-step explanation:

7 gallons = 210 miles

750-210= 540

210*2 =420 miles 7*2=14 gallons

210/2= 105 7/2=3.5 gallons

525 miles 17.5 miles

210/14= 15 7/14=.5 gallons

540 miles 18 gallons

6 0

2 years ago

It's not a college assignment!

Kisachek [45]

Using a calculator, the equation for the line of best fit where x represents the month and y represents the time is given by:

a. y = −1.74x + 46.6

<h3>How to find the equation of linear regression using a calculator?</h3>

To find the equation, we need to insert the points (x,y) in the calculator.

For this problem, the points (x,y) are given as follows, from the given table:

(1, 46), (2, 42), (3,40), (4, 41), (5, 38), (6,36).

Hence, inserting these points in the calculator, the equation for the line of best fit where x represents the month and y represents the time is given by:

a. y = −1.74x + 46.6

More can be learned about a line of best fit at brainly.com/question/22992800

#SPJ1

8 0

1 year ago

Find the circumference of a pizza with a diameter of 10 inches. Round to the nearest tenth.

olganol [36]

Answer:

The answer is 62.8

Step-by-step explanation:

C=2(3.14)(r)

c=6.28(10)

c=62.8

7 0

2 years ago

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Some advantages of consolidating your debt into a single loan are _________. Help Please !