PLS PLS PLS HELP Honors 9th Grade Math!!!!

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

4 years ago

Numbers like 10 100 1,000 and so are called what

sammy [17]

10 = Ten

100 = One Hundred

1,000 = One Thousand

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10 = 10^1

100 = 10^2 = 10 x 10

1,000 = 10^3 = 10 x 10 x 10

3 0

3 years ago

Read 2 more answers

What is 6 gallons in litres, to the nearest litre?

cestrela7 [59]

Answer:

22.71

Step-by-step explanation:

Please mark me branliest?!?!

5 0

3 years ago

What is

arsen [322]

9514 1404 393

Answer:

B. 17°

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you of the relations between angles and sides. Here, the relevant relation is ...

Sin = Opposite/Hypotenuse

sin(N) = PQ/PN

sin(N) = 4/14

N = arcsin(4/14) ≈ 16.6° . . . . use the inverse function to find the angle

N ≈ 17°

8 0

3 years ago

Type the standard form of three hundred thousand two hundred one

leonid [27]

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Standard form is numbers only.

Examples of standard numbers are 4, 225, or 9,002

A thousands number has 4 digits.

A hundreds number has 3 digits.

A tens number has 2 digits.

A regular number has 1 digit (itself).

Three hundred means 300.

Three hundred thousand, would be 300 times 1,000 (one thousand), or adding 3 zeros.

Two hundred is 200.

One is 1.

300 times 1,000 is 300,000.

300,000 and 200 is 300,200.

If the ones digit has a 1, then the number is 300,201.

Three hundred thousand, two hundred one written in standard form is 300,201.

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5 0

3 years ago

Read 2 more answers