-|7(-5)+7(-8)|-|-6(-7)|

Write down the explicit solution for each of the following: a) x’=t–sin(t); x(0)=1

Kay [80]

Answer:

a) x=(t^2)/2+cos(t), b) x=2+3e^(-2t), c) x=(1/2)sin(2t)

Step-by-step explanation:

Let's solve by separating variables:

$x'=\frac{dx}{dt}$

a) x’=t–sin(t), x(0)=1

$dx=(t-sint)dt$

Apply integral both sides:

$\int {} \, dx=\int {(t-sint)} \, dt\\\\x=\frac{t^2}{2}+cost +k$

where k is a constant due to integration. With x(0)=1, substitute:

$1=0+cos0+k\\\\1=1+k\\k=0$

Finally:

$x=\frac{t^2}{2} +cos(t)$

b) x’+2x=4; x(0)=5

$dx=(4-2x)dt\\\\\frac{dx}{4-2x}=dt \\\\\int {\frac{dx}{4-2x}}= \int {dt}\\$

Completing the integral:

$-\frac{1}{2} \int{\frac{(-2)dx}{4-2x}}= \int {dt}$

Solving the operator:

$-\frac{1}{2}ln(4-2x)=t+k$

Using algebra, it becomes explicit:

$x=2+ke^{-2t}$

With x(0)=5, substitute:

$5=2+ke^{-2(0)}=2+k(1)\\\\k=3$

Finally:

$x=2+3e^{-2t}$

c) x’’+4x=0; x(0)=0; x’(0)=1

Let $x=e^{mt}$ be the solution for the equation, then:

$x'=me^{mt}\\x''=m^{2}e^{mt}$

Substituting these equations in c)

$m^{2}e^{mt}+4(e^{mt})=0\\\\m^{2}+4=0\\\\m^{2}=-4\\\\m=2i$

This becomes the solution m=α±βi where α=0 and β=2

$x=e^{\alpha t}[Asin\beta t+Bcos\beta t]\\\\x=e^{0}[Asin((2)t)+Bcos((2)t)]\\\\x=Asin((2)t)+Bcos((2)t)$

Where A and B are constants. With x(0)=0; x’(0)=1:

$x=Asin(2t)+Bcos(2t)\\\\x'=2Acos(2t)-2Bsin(2t)\\\\0=Asin(2(0))+Bcos(2(0))\\\\0=0+B(1)\\\\B=0\\\\1=2Acos(2(0))\\\\1=2A\\\\A=\frac{1}{2}$

Finally:

$x=\frac{1}{2} sin(2t)$

7 0

3 years ago

In the figure shown below, AC is parallel to DE with transversal BF

densk [106]

Answer:

Given -  AC is parallel to DE

angle 3y + 28 = angle X ( Vertically opposite angles are equal).

3y + 28 = X

X + Y = 180° ( supplementary angles).

3y - X = (-28)

Solving both the equation we get

multiply eq 1 with 3

3x + 3y = 540

-X + 3y = (-28)

subtracting

4x = 568

x = 142°

3y + 28 = 142

3y = 142 - 28

3y = 114

y = 38°

4 0

2 years ago

Why is the percent increase from 45 to 75 not equal to the percent decrease from 75 to 45? Select three

Novay_Z [31]

Answer:

For me the answer is

The ratio for the percent increase has a smaller denominator than the percent decrease.

Step-by-step explanation:

From 45 to 75, the increase is 30

If we check the % increased, (30/45)*100 = 66.667%

If we decrease the price from 75 to 45, the amount is same, i.e. 30 but the denominator is changed.

(30/75)*100 = 40%

5 0

2 years ago

Compute the discriminant. Then determine the number and type of solutions for the given equation.

Neporo4naja [7]

Answer:

Step-by-step explanation:

It's two unequal real solutions.

a = 1

b = 7

c = 6

discriminate = sqrt(7^2 - 4*1*6)

discriminate = sqrt(49 - 24)

discriminate = sqrt(25)

discriminate = 5

Now you cannot determine what this means unless you put it in the quadratic formula

x = (-7 +/- 5)/2

x = (- 7 + 5) / 2

x = - 2/2 = - 1

x = (-7 - 5)/2

x = -12/2

x = - 6

You see that you have two unequal reals.

7 0

2 years ago

Help. I’m stuck on this one skskksks

VMariaS [17]

I'm pretty sure the answer is: h = 9

Reason:

The question tells us that to complete the equation that represents the relationship between d and h, so for "d", its d = 1, for h, it is 9 per each "days"

from the hours.

6 0

3 years ago

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