1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
bearhunter [10]
3 years ago
8

Find a solution to the initial value problem, y′′+18x=0,y(0)=5,y′(0)=1.

Mathematics
1 answer:
Serga [27]3 years ago
7 0

We want to find a solution to the initial value problem:

y'' + 18x = 0 \qquad,\qquad y(0) = 5 \qquad,\qquad y'(0)=1.

We can start by integrating the equation once:

\dfrac{\textrm{d}^2 y}{\textrm{d}x^2} + 18 x = 0 \iff \dfrac{\textrm{d}^2 y}{\textrm{d}x^2} = -18 x \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -18\displaystyle\int x\textrm{ d}x \iff \dfrac{\textrm{d}y}{\textrm{d}x}=-18\dfrac{x^2}{2} + C \iff\\\\\iff \dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + C.

Using the initial condition y'(0) = 1, we can determine the integration constant C:

\dfrac{\textrm{d}y}{\textrm{d}x}\Big\vert_{x= 0} = 1 \iff -9 \times 0^2 + C = 1 \iff C = 1.

Therefore, we have:

\dfrac{\textrm{d}y}{\textrm{d}x} = -9x^2 + 1

We can now integrate again:

y(x) = \displaystyle\int\dfrac{\textrm{d}y}{\textrm{d}x}\textrm{ d}x = \int\left(-9x^2+1\right)\textrm{d}x = -9\int x^2\textrm{ d}x + \int\textrm{d}x =\\\\= -9\dfrac{x^3}{3} + x + K = -3x^3 + x + K.

The integration constant K is determined by using y(0) = 5:

y(0) = 5 \iff -3 \times 0^3 + 0 + K = 5 \iff K = 5.

Finally, the solution is:

\boxed{y(x) = -3x^3 + x + 5}.

You might be interested in
Phil has budgeted 120 to rent a car during his weeklong vacation. At the rate of $40 per week plus 10 cents per mile how many mi
Elanso [62]

800 miles     120-40=80 so he has $80 worth of miles to use. He can go 10 miles on $1 so 80*10=800

He can go 800 miles.

4 0
2 years ago
4(x-3)=5(x+1) solve the equation above for x
lesantik [10]
First distribute: 4x-12=5x+5. subtract 4x on both sides: -12=x+5. Now subtract 5 from both sides. x=-17. 
6 0
3 years ago
Company A offers unlimited minutes and hacks for a flat fee of $50 per month and charges $1.50 for each GB of data used per mont
GREYUIT [131]
F(x) = 1.5x + 50

if you use 5GB per month, you should pay
f(5) = 1.5(5) + 50 = $57.50
4 0
3 years ago
Marsha wants to determine the vertex of the quadratic function f(x) = x2 – x 2. What is the function’s vertex? (one-half, seven-
nydimaria [60]

Vertex of the quadratic equation is the highest or the lowest point of the quadratic equation.

<h3>Given-</h3>

The given equation in the question is,

f(x) = x^2 - x +2

<h3>Vertex of the quadratic equation</h3>

Vertex of the quadratic equation is the highest or the lowest point of the quadratic equation.

To find the vertex of the equation we need to compare the given equation with the following equation.

f(x)=a(x-h)^2+k

Here h and k are the vertex of the equation.

The given equation is,

f(x) = x^2 - x +2

Rewrite the equation,

f(x) = x^2 - x +(\dfrac{1}2)^2} +2-(\dfrac{1}2)^2}

f(x) = x^2 - x +\dfrac{1}{4} +2-\dfrac{1}4}

f(x) =(x-\dfrac{1}{2} )(x-\dfrac{1}{2} )+2-\dfrac{1}{4}

f(x) =(x-\dfrac{1}{2} )^2+\dfrac{7}{4}

Thus the vertex of the quadratic equation is 1/2 and 7/4.

Learn more about the vertex of the quadratic equation here;

brainly.com/question/6356924

3 0
2 years ago
The president of a small manufacturing firm is concerned about the continual increase in manufacturing costs over the past sever
mrs_skeptik [129]

Answer:

The answers to the question are

(a) The time series plot is given as attached

(b) The parameters for the line that minimizes MSE for the time series are;

y-intercept, b₀ = 19.993

Slope, b₁ =  1.77

MSE = 19.44

T_t = 19.993  + 1.774·t

(c) The average cost increase that the firm is realizing per year is $ 1.77

(d) The estimate of the cost/unit for next year is $35.96.

Step-by-step explanation:

(a) Using the provided data, the time series plot is given as attached

(b) Given hat the y-intercept, = b₀

Slope = b₁

Therefore the  linear trend forecast equation is given s

T_t = b₀ + b₁·t

The linear trend line slope is given as

b₁ = \frac{\Sigma^n_{t=1}(t-\overline{\rm t)}(Y_t-\overline{\rm Y)}}{\Sigma^n_{t=1}(t-\overline{\rm t} )^2}

b₀ = \overline{\rm Y} - b₁·\overline{\rm t}

Where:

Y_t = Time series plot value at t

n =  Time period number

\overline{\rm Y} = Time series data average value and

\overline{\rm t} = Average time, t

Therefore, \overline{\rm t} = \frac{\Sigma^n _{t=1} t}{n}  = \frac{36}{8} =4.5

\overline{\rm t} = 4.5

\overline{\rm Y}  =  \frac{\Sigma^n _{t=1} Y_t}{n}  = \frac{223.8}{8} =27.975

\overline{\rm Y}  =  27.975

Therefore the linear trend line equation T_t, is

b₁ = \frac{\Sigma^n_{t=1}(t-\overline{\rm t)}(Y_t-\overline{\rm Y)}}{\Sigma^n_{t=1}(t-\overline{\rm t} )^2} =  = \frac{74.5}{42} = 1.774

b₀ = \overline{\rm Y} - b₁·\overline{\rm t} = 27.975 - 1.774×4.5 = 19.993

Therefore the trend equation for the linear trend is

T_t = 19.993  + 1.774·t

MSE = \frac{1}{2} \Sigma(Y-\overline{\rm Y)^ }^2 = \frac{155.495}{8} = 19.44

(c) From the linear trend equation, the average is given as the slope of the curve or b₁ which is equal to 1.774

Therefore the average cost increase that the firm has been realizing per year is $ 1.77

(b) From the equation of the future trend, we have when y = 9

T_t  is given as

T_t = 19.993  + 1.774×9 = 35.96

The cost/unit for 9th year is $35.96

3 0
3 years ago
Other questions:
  • A family eats at a restaurant that is having a 15% discount promotion. THeir meal cost $78.65, and they leave a 20% tip. If the
    5·1 answer
  • A school orders 12 boxes of dice. Each box contains six bags of dice. Each bag contains
    13·2 answers
  • A government that is ruled by the people is a __________. A. republic B. monarchy C. democracy D. dictatorship
    6·2 answers
  • A copy machine makes 32 copies per minute. How long does it take to make 120 copies?
    11·2 answers
  • The following number sentence is _____________.
    13·1 answer
  • A birthday hat is the same shape as a *<br><br> triangle<br> cone<br> ice cream scoop
    11·1 answer
  • If h=11 units and r =2 units then what is the approximate volume of the cone
    14·1 answer
  • The number of veggie burgers sold at a restaurant in Houston, Texas, was 425 in April. The number sold in May was 16% less than
    11·2 answers
  • Rose wants to buy a toy that costs $14. If she gives the cashier $20, which equation will tell you how much change Rose gets bac
    15·1 answer
  • Great White Sharks can grow up to 36 feet in length. Suppose a shark is
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!