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
alekssr [168]
3 years ago
10

Consider the BVP

Mathematics
1 answer:
babymother [125]3 years ago
6 0

y''+\lambda y=0

has the corresponding characteristic equation (CE)

r^2+\lambda=0

a. If \lambda=0, then the CE has one root, r=0, and so the general solution to the ODE is

y(t)=C_1+C_2t

Given that y'(0)=y'\left(\frac\pi6\right)=0, and

y'(t)=C_2

it follows that C_2=0, and so

\boxed{y(t)=C_1}

b. If \lambda>0, then the CE has two complex roots, r=\pm i\sqrt\lambda, and the general solution is

y(t)=C_1\cos(\lambda t)+C_2\sin(\lambda t)

\implies y'(t)=-\lambda C_1\sin(\lambda t)+\lambda C_2\cos(\lambda t)

With the given boundary values, we have

y'(0)=0\implies\lambda C_2=0\implies C_2=0

y'\left(\dfrac\pi6\right)=0\implies-\lambda C_1\sin\left(\dfrac{\lambda\pi}6\right)=0

\implies\sin\left(\dfrac{\lambda\pi}6\right)=0

\implies\dfrac{\lambda\pi}6=n\pi

\implies\lambda=6n

where n\in\Bbb Z.

  • If \lambda is a (positive) multiple of 6, we have

y'\left(\dfrac\pi6\right)=0\implies-6nC_1\sin\left(\dfrac{6n\pi}6\right)=0\implies C_1=0

and the solution would be

\boxed{y(t)=0}

  • Otherwise, if \lambda is not a multiple of 6, we have

y'\left(\dfrac\pi6\right)=0\implies-\lambda C_1\sin\left(\dfrac{\lambda\pi}6\right)=0\implies C_1=0

so that we still get

\boxed{y(t)=0}

c. If \lambda, then the CE has two real roots, r=\pm\sqrt\lambda, so that the general solution is

y(t)=C_1e^{\sqrt\lambda\,t}+C_2e^{-\sqrt\lambda\,t}

\implies y'(t)=C_1\sqrt\lambda\,e^{\sqrt\lambda\,t}-C_2\sqrt\lambda\,e^{-\sqrt\lambda\,t}

From the boundary conditions we get

y'(0)=0\implies C_1\sqrt\lambda-C_2\sqrt\lambda=0\implies C_1=C_2

y'\left(\dfrac\pi6\right)=0\implies C_1\sqrt\lambda\,e^{(\pi\sqrt\lambda)/6}-C_2\sqrt\lambda\,e^{-(\pi\sqrt\lambda)/6}=0\implies C_1e^{(\pi\sqrt\lambda)/3}=C_2

from which it follows that C_1=C_2=0, so again the solution is

\boxed{y(t)=0}

d. We only get eigenvalues in the case when \lambda>0, as in part (b):

\boxed{\lambda=6n,\,n\in\{1,2,3,\ldots\}}

for which we get the corresponding eigenfunctions

\boxed{y(t)=\cos(6nt),\,n\in\{1,2,3,\ldots\}}

You might be interested in
Write a polynomial f (x) that satisfies the given conditions. Polynomial of lowest degree with zeros of -4 (multiplicity 3), 1 (
yuradex [85]

Answer:

Step-by-step explanation:

Polynomial f(x) has the following conditions: zeros of -4 (multiplicity 3), 1 (multiplicity 1), and with f(0) = 320.

The first part zeros of -4 means (x+4) and multiplicity 3 means (x+4)^3.

The second part zeros of 1 means (x-1) and multiplicity 1 means (x-1).

The third part f(0) = 320 means substituting x=0 into (x+4)^3*(x-1)*k =320

(0+4)^3*(0-1)*k = 320

-64k = 320

k = -5

Combining all three conditions, f(x)

= -5(x+4)^3*(x-1)

= -5(x^3 + 3*4*x^2 + 3*4*4*x + 4^3)(x-1)

= -5(x^4 + 12x^3 + 48x^2 + 64x - x^3 - 12x^2 - 48x - 64)

= -5(x^4 + 11x^3 + 36x^2 + 16x -64)

= -5x^3 -55x^3 - 180x^2 - 80x + 320

8 0
2 years ago
Read 2 more answers
Jessa had 23 one dollar bills that she wants to divide equally between her 5 children. How much money will each receive? How muc
vazorg [7]

Answer:

$4.06

Step-by-step explanation:

23 / 5 = 4

Remainder = 3

$3 into dimes = 30 dimes

30 / 5 = 6

$4 + €6 = $4.06

6 0
3 years ago
Directions - For the following problem, write a paragraph proof to justify each step you make. All work must be neat,
nlexa [21]

$ PS = \frac{4}{3} x

Solution:

Given PRQ is a triangle.

ST is a line parallel to RQ.

$PT = x, \ PQ = 3x,  \ SR=\frac{8}{3}x

TQ=PQ-PT

TQ=3x -x=2x

<u>Triangle proportionality theorem,</u>

<em>If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.</em>

$\frac{PS}{SR} =\frac{PT}{TQ}

$\frac{PS}{\frac{8}{3} x} =\frac{x}{2x}

Do cross multiplication, we get

$ PS \times 2x=x \times \frac{8}{3} x

Divide by 2x on both sides, we get

$ PS = \frac{4}{3} x

4 0
3 years ago
A plane flies with an average velocity of -98.5 m/s for 45.0s .What is the displacement?
Gnesinka [82]

Answer:

  -4432.5 m

Step-by-step explanation:

Displacement is measured in meters, so will be the product of velocity in m/s and time in s.

  (-98.5 m/s)×(45.0 s) = -4432.5 m

___

If you're concerned with significant figures, you can round this to -4430 m, which has the required 3 significant figures.

8 0
3 years ago
Read 2 more answers
3. Grad point question for a quiz<br><br>Graph the inequality.<br><br>A. x -3<br>D. x≤-3
LUCKY_DIMON [66]

Answer:

The answer most likely would be x > -3

Step-by-step explanation:

It would be this because it's a opened dot and not a closed one. A closed dot would have the greater or less than sign with the line underneath it

REMEMBER if it is shaded to the right then the inequality points to the right (>).  If it is shaded to the left the inequality points to the left (<)

HOPE THIS HELPS

3 0
3 years ago
Other questions:
  • A density curve for all the possible weights between 0 pounds and 10 pounds is in the shape of a rectangle. What is the value of
    8·2 answers
  • An ice bucket is composed of a cylinder and a hemisphere, as shown. Crushed ice weighs 0.34 ounce per cubic inch. What is the ap
    6·1 answer
  • The length of a rectangle is 13 feet. The width is 9 feet. What is the perimeter of the rectangle?
    14·2 answers
  • Half of your baseball card collection got wet and was ruined. you bought 14 cards to replace some that were lost.how many did yo
    9·1 answer
  • Find the product of 100 abc(0.01a - 0.019bc)
    5·1 answer
  • In factons you divide the numerator and the whole number .. then denominator<br><br> Correct?
    8·1 answer
  • Solve for x. 9^6 = x^3 • 9^3
    15·1 answer
  • If a quadrilateral has angles measure 2y,3y ,y+5and 2y-1,what is the value of y
    7·1 answer
  • PLEASE HELP 30 POINTS <br> Find the angle measure indicated.<br> 140 °<br> ?
    14·1 answer
  • Solve for the following equation<br>6Y+4 = 4y-2​
    8·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!