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
son4ous [18]
2 years ago
12

Help with q25 please. Thanks.​

Mathematics
1 answer:
Westkost [7]2 years ago
7 0

First, I'll make f(x) = sin(px) + cos(px) because this expression shows up quite a lot, and such a substitution makes life a bit easier for us.

Let's apply the first derivative of this f(x) function.

f(x) = \sin(px)+\cos(px)\\\\f'(x) = \frac{d}{dx}[f(x)]\\\\f'(x) = \frac{d}{dx}[\sin(px)+\cos(px)]\\\\f'(x) = \frac{d}{dx}[\sin(px)]+\frac{d}{dx}[\cos(px)]\\\\f'(x) = p\cos(px)-p\sin(px)\\\\ f'(x) = p(\cos(px)-\sin(px))\\\\

Now apply the derivative to that to get the second derivative

f''(x) = \frac{d}{dx}[f'(x)]\\\\f''(x) = \frac{d}{dx}[p(\cos(px)-\sin(px))]\\\\ f''(x) = p*\left(\frac{d}{dx}[\cos(px)]-\frac{d}{dx}[\sin(px)]\right)\\\\ f''(x) = p*\left(-p\sin(px)-p\cos(px)\right)\\\\ f''(x) = -p^2*\left(\sin(px)+\cos(px)\right)\\\\ f''(x) = -p^2*f(x)\\\\

We can see that f '' (x) is just a scalar multiple of f(x). That multiple of course being -p^2.

Keep in mind that we haven't actually found dy/dx yet, or its second derivative counterpart either.

-----------------------------------

Let's compute dy/dx. We'll use f(x) as defined earlier.

y = \ln\left(\sin(px)+\cos(px)\right)\\\\y = \ln\left(f(x)\right)\\\\\frac{dy}{dx} = \frac{d}{dx}\left[y\right]\\\\\frac{dy}{dx} = \frac{d}{dx}\left[\ln\left(f(x)\right)\right]\\\\\frac{dy}{dx} = \frac{1}{f(x)}*\frac{d}{dx}\left[f(x)\right]\\\\\frac{dy}{dx} = \frac{f'(x)}{f(x)}\\\\

Use the chain rule here.

There's no need to plug in the expressions f(x) or f ' (x) as you'll see in the last section below.

Now use the quotient rule to find the second derivative of y

\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right]\\\\\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{f'(x)}{f(x)}\right]\\\\\frac{d^2y}{dx^2} = \frac{f''(x)*f(x)-f'(x)*f'(x)}{(f(x))^2}\\\\\frac{d^2y}{dx^2} = \frac{f''(x)*f(x)-(f'(x))^2}{(f(x))^2}\\\\

If you need a refresher on the quotient rule, then

\frac{d}{dx}\left[\frac{P}{Q}\right] = \frac{P'*Q - P*Q'}{Q^2}\\\\

where P and Q are functions of x.

-----------------------------------

This then means

\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + p^2\\\\\frac{f''(x)*f(x)-(f'(x))^2}{(f(x))^2} + \left(\frac{f'(x)}{f(x)}\right)^2 + p^2\\\\\frac{f''(x)*f(x)-(f'(x))^2}{(f(x))^2} +\frac{(f'(x))^2}{(f(x))^2} + p^2\\\\\frac{f''(x)*f(x)-(f'(x))^2+(f'(x))^2}{(f(x))^2} + p^2\\\\\frac{f''(x)*f(x)}{(f(x))^2} + p^2\\\\

Note the cancellation of -(f ' (x))^2 with (f ' (x))^2

------------------------------------

Let's then replace f '' (x) with -p^2*f(x)

This allows us to form  ( f(x) )^2 in the numerator to cancel out with the denominator.

\frac{f''(x)*f(x)}{(f(x))^2} + p^2\\\\\frac{-p^2*f(x)*f(x)}{(f(x))^2} + p^2\\\\\frac{-p^2*(f(x))^2}{(f(x))^2} + p^2\\\\-p^2 + p^2\\\\0\\\\

So this concludes the proof that \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + p^2 = 0\\\\ when y = \ln\left(\sin(px)+\cos(px)\right)\\\\

Side note: This is an example of showing that the given y function is a solution to the given second order linear differential equation.

You might be interested in
Gretchen has a set of blocks of heights 1, 2, and 4-centimeters. Imagine stacking the blocks one on top of the other to make a t
notsponge [240]

Answer:

t_{n}=t_{n-1}+t_{n-2} +t_{n-4}

Step-by-step explanation:

t_{n}=multiple ways to climb a tower

When n = 1,

tower= 1 cm  

t_{1}= 1

When n = 2,

tower =2 cm   

t_{2}= 2

When n = 3,

tower = 3 cm

it can be build if we use three 1 cm blocks

t_{3} = 3

When n = 4,

tower= 4 cm

it can be build if we use four 1 cm blocks

t_{4} = 6

When n > 5

tower height > 4 cm

so we can use 1 cm, 2 cm and 4 cm blocks

so in that case if our last move is 1 cm block then t_{n-1} will be

n —1 cm

if our last move is 2 cm block then t_{n-2} will be

n —2 cm

if our last move is 4 cm block then t_{n-4} will be

n —4 cm

 

t_{n}=t_{n-1}+t_{n-2} +t_{n-4}

4 0
3 years ago
Read 2 more answers
Which of the following is the smallest set? Becareful of spelling. Real numbers, Rational Numbers, Irrational numbers or Natural
sattari [20]
Real because it’s 12345678910
6 0
3 years ago
2x^2-5x+6 divided by x-3
hichkok12 [17]

Answer:

2

Step-by-step explanation:


8 0
2 years ago
Which table represents a linear function?
bearhunter [10]

Answer:

  • Table 2

Step-by-step explanation:

<h3>Table 1</h3>
  • x- values change 1 to 4
  • y- values change inconsistently and repeat at -2

This is <u>not</u> a linear function

<h3>Table 2</h3>
  • x- values change 1 to 4
  • y- values change consistently, with common difference of -2

<u>This is a linear function</u>

<h3>Table 3</h3>
  • x- values change 1 to 4
  • y- values change inconsistently, the difference is not common

This is <u>not</u> a linear function

<h3>Table 4</h3>
  • x- values change 1 to 4
  • y- values change inconsistently, the difference is not common

This is <u>not</u> a linear function

8 0
2 years ago
For one child, a childcare facility charges $350 per month for preschool and $5.00 per hour for each additional hour for after-s
Hatshy [7]

Answer:

D. 80

Step-by-step explanation:

We have the following function  :

f(h)=350+5h (I)

Where f(h) is the monthly fee for after - school care and preschool.

Where h represents the number of hours spent in after - school care.

We know that the total monthly bill for one child was $750 ⇒

f(h)=750

Using (I) we will find the value of the hours ''h'' that verifies the expression ⇒

f(h)=750=350+5h ⇒

750=350+5h\\750-350=5h\\400=5h\\h=\frac{400}{5}=80

We find that the value of the hours ''h'' is 80 hours.

The total number of hours the child spent in after - school care is D. 80

5 0
3 years ago
Other questions:
  • Which of the following items has the lowest unit price? 4 for $5.00 $1.22 each 6 for $7.44 3 for $3.60
    7·1 answer
  • I need help with this!!!!!!
    10·2 answers
  • How many units up was the graph of f(x) =∛x shifted to form the translation?
    8·2 answers
  • the length of a rectangle is 2 centimeters more than the width the perimeter is 20 find the length and width
    14·1 answer
  • I really don’t understand this question, please help.
    7·2 answers
  • Find the 26th term of the following sequence:<br> 4, 0, -4, -8...
    13·1 answer
  • Please help!!! Will give brainliest!
    6·2 answers
  • Given mn, find the value of x.<br> 108
    13·1 answer
  • 17 cm<br> 12 cm<br> 5 cm<br> pls help
    5·1 answer
  • Jamie sees a sale at the Gaming Store where all games are the same price. He buys 3 video games. His sister talks him into also
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!