. Write a number in which the value of<br> the 6 is one tenth the value of the 6 ir<br> 35,629.

mars1129 [50]

Answer:

$y=e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+\frac{1}{\sqrt{3}}\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )$

Step-by-step explanation:

A second order linear , homogeneous ordinary differential equation has form $ay''+by'+cy=0$ .

Given: $y''+y'+y=0$

Let $y=e^{rt}$ be it's solution.

We get,

$\left ( r^2+r+1 \right )e^{rt}=0$

Since $e^{rt}\neq 0$ , $r^2+r+1=0$

{ we know that for equation $ax^2+bx+c=0$ , roots are of form $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ }

We get,

$y=\frac{-1\pm \sqrt{1^2-4}}{2}=\frac{-1\pm \sqrt{3}i}{2}$

For two complex roots $r_1=\alpha +i\beta \,,\,r_2=\alpha -i\beta$ , the general solution is of form $y=e^{\alpha t}\left ( c_1\cos \beta t+c_2\sin \beta t \right )$

i.e $y=e^{\frac{-t}{2}}\left ( c_1\cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )$

Applying conditions y(0)=1 on $e^{\frac{-t}{2}}\left ( c_1\cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )$ , $c_1=1$

So, equation becomes $y=e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )$

On differentiating with respect to t, we get

$y'=\frac{-1}{2}e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+c_2\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )+e^{\frac{-t}{2}}\left ( \frac{-\sqrt{3}}{2} \sin \left ( \frac{\sqrt{3}t}{2} \right )+c_2\frac{\sqrt{3}}{2}\cos\left ( \frac{\sqrt{3}t}{2} \right )\right )$

Applying condition: y'(0)=0, we get $0=\frac{-1}{2}+\frac{\sqrt{3}}{2}c_2\Rightarrow c_2=\frac{1}{\sqrt{3}}$

Therefore,

$y=e^{\frac{-t}{2}}\left ( \cos\left ( \frac{\sqrt{3}t}{2} \right )+\frac{1}{\sqrt{3}}\sin \left ( \frac{\sqrt{3}t}{2} \right ) \right )$

3 0

2 years ago

What is the range of the given function

Lelechka [254]

Domain is X and Range is Y. So the range will be all the Y's which is the second answer

7 0

2 years ago

Read 2 more answers

carson walks his dog 6 blocks south to the park. Next they jog 5 blocks east to go swimming at the pond. How many blocks did car

Vinil7 [7]

Pretty simple, 5 + 6 = 11 blocks total

4 0

2 years ago

In 1996, funding for a program increased by 0.62 billion dollars from the funding in 1995. In 1997, the funding increased 0.29 b

11111nata11111 [884]

1995 = x
1996 = x + 0.62
1997 = x + 0.62 + 0.29

x + x + 0.62 + x + 0.62 + 0.29 = 11.25
3x + 1.53 = 11.25
3x = 11.25 - 1.53
3x = 9.72
x = 9.72 / 3
x = 3.24 <== 1995 = 3.24 million

1996 = 3.24 + 0.62 = 3.86 million <==
1997 = 3.24 + 0.62 + 0.29 = 4.15 million <==

4 0

2 years ago

Raul works at a movie theatre. The function f(x) represents the amount of money Raul earns per ticket, where x is the number of

kipiarov [429]

Step-by-step explanation:

Let's call A is the number of tickets Raul sells per hour.

g(x) = A*x

Let's call B is the amount of money Raul earns per ticket.

f(g(x)) = B*g(x) = B*A*x

4 0

1 year ago

. Write a number in which the value of the 6 is one tenth the value of the 6 ir 35,629.