Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

et
x(t) be the solution of the initial value problem:

=-x-x^(3),x(0)=0,x^(')(0)=2". "
Vhen
x=1, the value of
4(x^(˙))^(2) is
qquad
ormula:
x^(¨)=(d)/(dx)((((x^(˙)))^(2))/(2))

et $x(t)$ be the solution of the initial value problem: $\newline$ $=-x-x^{3}, x(0)=0, x^{\prime}(0)=2 \text {. }$ $\newline$ Vhen $x=1$ , the value of $4(\dot{x})^{2}$ is $\qquad$ $\newline$ ormula: $\ddot{x}=\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{(\dot{x})^{2}}{2}\right)$

View step-by-step help

View step-by-step help

Write equations of cosine functions using properties

Full solution

Q. et

x(t)

be the solution of the initial value problem:

\newline

=-x-x^{3}, x(0)=0, x^{\prime}(0)=2 \text {. }

\newline

Vhen

x=1

, the value of

4(\dot{x})^{2}

is

\qquad

\newline

ormula:

\ddot{x}=\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{(\dot{x})^{2}}{2}\right)

Given Differential Equation: Given the differential equation: x^{\¨} = -x - x^{3} and initial conditions $x(0) = 0$ , $x^{\prime}(0) = 2$ .
Differentiate Velocity: To find x^{\¨}, we differentiate the velocity x^{\˙} with respect to $x$ using the chain rule: x^{\¨} = \frac{d}{dx}\left(\frac{(x^{\˙})^2}{2}\right).
Set Equation Equal: We know that x^{\¨} = -x - x^{3}. So, we set (d)/(dx)((x^{\˙})^2/2) = -x - x^{3}.
Integrate Both Sides: Multiply both sides by $2$ to get rid of the fraction: $\frac{d}{dx}(x^{\dot{}})^2$ = $-2x - 2x^{3}$ .
Find Integration Constant: We need to find $(x^{\dot{}})^2$ when $x=1$ . Integrate both sides with respect to $x$ to find $(x^{\dot{}})^2$ as a function of $x$ .
Use Initial Condition: The integral of the right side with respect to $x$ is $-x^2 - \frac{1}{2}x^4 + C$ , where $C$ is the integration constant.
Calculate Integration Constant: To find $C$ , use the initial condition $x'(0) = 2$ , which means $(x^{\dot{}})^2 = 4$ when $x = 0$ .
Find Velocity Squared: Plug $x = 0$ into $-x^2 - \frac{1}{2}x^4 + C$ to get $C$ : $4 = -0^2 - \frac{1}{2}0^4 + C$ , so $C = 4$ .
Calculate Velocity Squared: Now we have $(\dot{x})^2 = -x^2 - \frac{1}{2}x^4 + 4$ . Plug in $x = 1$ to find $(\dot{x})^2$ : $(\dot{x})^2 = -1^2 - \frac{1}{2}1^4 + 4$ .
Multiply Velocity Squared: Calculate $(x^{(\dot{})})^2$ when $x = 1$ : $(x^{(\dot{})})^2 = -1 - \left(\frac{1}{2}\right)(1) + 4$ .
Multiply Velocity Squared: Calculate $(x^{(\dot{})})^2$ when $x = 1$ : $(x^{(\dot{})})^2 = -1 - (\frac{1}{2})(1) + 4$ . Simplify the expression: $(x^{(\dot{})})^2 = -1 - \frac{1}{2} + 4 = 2.5$ .
Multiply Velocity Squared: Calculate $(x^{(\dot{})})^2$ when $x = 1$ : $(x^{(\dot{})})^2 = -1 - (\frac{1}{2})(1) + 4$ . Simplify the expression: $(x^{(\dot{})})^2 = -1 - \frac{1}{2} + 4 = 2.5$ . Finally, multiply $(x^{(\dot{})})^2$ by $4$ to find $4(x^{(\dot{})})^2$ : $4(x^{(\dot{})})^2 = 4 \times 2.5 = 10$ .

More problems from Write equations of cosine functions using properties

Question

Find an equation for a sinusoidal function that has period

2\pi

1

, and contains the point

\left(\pi, -1\right)

\newline

Write your answer in the form

f(x)=A\cos(Bx+C)+D

A

B

C

D

are real numbers.

\newline

f(x)=

Posted 1 month ago

Question

Find an equation for a sinusoidal function that has period

2\pi

1

, and contains the point

\left(-\frac{\pi}{2},-1\right)

. Write your answer in the form

f(x) = A\sin(Bx + C) + D

A

B

C

D

are real numbers. `f(x)` =______

Posted 3 months ago

Question

Find an equation for a sinusoidal function that has period

2\pi

1

, and contains the point

(0,-1)

\newline

Write your answer in the form

f(x) = A\cos(Bx + C) + D

A

B

C

D

are real numbers.

\newline

f(x) =

Posted 2 months ago

Question

Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought

10

cans of food, and each student brought

2

cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought

3

cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class

y

in terms of the number of students

x

Posted 2 months ago

Question

Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought

10

cans of food, and each student brought

2

cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought

3

cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class

y

in terms of the number of students

x

Posted 2 months ago

Question

Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is

12

feet long, and each strand of his Christmes lights will wrap around a

70

-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?

\newline

6

\newline

7

\newline

8

\newline

9

Posted 2 months ago

Question

Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is

12

feet long, and each strand of his Christmas lights will wrap around a

70

-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?

\newline

6

\newline

7

\newline

8

\newline

9

Posted 2 months ago

Question

For a given input value

b

f

outputs a value

a

to satisfy the following equation.

\newline

4a+7b=-52

\newline

Write a formula for

f(b)

b

\newline

f(b)=\square

Posted 2 months ago

Question

For a given input value

x

g

outputs a value

y

to satisfy the following equation.

\newline

-4x-6=-5y+2

\newline

Write a formula for

g(x)

x

\newline

g(x)=\square

Posted 4 months ago

Question

For a given input value

a

f

outputs a value

b

to satisfy the following equation.

\newline

-3a+6b=a+4b

\newline

Write a formula for

f(a)

a

\newline

f(a)=\square

Posted 4 months ago