Whoever gets first gets brainiest 26 times 98

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

3 years ago

Kwame bought a car for $25,000. The value of the car is decreasing at a rate of 6.5% each year. Write a function that represents

sweet-ann [11.9K]

<h3>The function that represents the price P of the car after x years is:</h3>

$P = 25000(0.935)^x$

Solution:

The decreasing function is given as:

$y = a(1 - r)^t$

Where,

y is future value

a is initial value

r is decreasing rate in decimal

t is time period

From given,

a = 25000

$r = 6.5 \% = \frac{6.5}{100} = 0.065$

number of years = x

future value = P

Therefore,

$P = 25000(1 - 0.065)^x\\\\P = 25000(0.935)^x$

Thus the function is found

8 0

4 years ago

What is the area of a rectangle with length 112 foot and width 34 foot?

Nastasia [14]

Answer:

3808ft²

Step-by-step explanation:

Formula: A=L×W

A= area

L= length

W= width

A= 112 × 34= 3808ft²

8 0

3 years ago

Which one is true!!!

irga5000 [103]

B and C I believe .............

7 0

3 years ago

In which number does the digit 4 have a value that is \dfrac{1}{10} 10 1 start fraction, 1, divided by, 10, end fraction times

MrRissso [65]

Answer:

The digit 4 should be in the thousand position in the number sought;

Examples

9.304,

9.004

Step-by-step explanation:

Here, we are required to consider decimal places of numbers

The digit 4 in the number 3.8463 is in the hundredth position

That is 0.04

We are asked to look for the number with a digit 4 that has the same value as $\frac{1}{10} \times$ the digit 4 in 3.8463

That is

$\frac{1}{10} \times 0.04 = 0.004$

Therefore, the digit 4 in the number sought should be in the thousandth position, that is 0.004

Example includes 9.304 or 9.004.

4 0

3 years ago