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
you have an initial amount of 55$ in a savings account and you deposit an equal amount into your account each week. After 6 week
sveticcg [70]
55+added=133
added=amount addeed per week times 6=6x

55+6x=133
minus 55 both sides
6x=78
divide both sides by 6
x=13

add 13 per week

y=13x+55
3 0
3 years ago
I need help lol last one
Andrei [34K]

Answer:

x= 19

If you need help with more, I'm more than happy to help you.

6 0
3 years ago
Read 2 more answers
What is the probability that the first marbles you choose is red and the second is blue
ella [17]

Answer:

2/9

Step-by-step explanation:

5 0
1 year ago
Which property is shown? If m ABC = m CBD, then m CBD = m ABC reflexive property substitution property symmetric property transi
trapecia [35]
The correct answer for the question that is being presented above is this one: "symmetric property." Symmetric Property has the property of a = b, then b = a. This property is one of the equivalent properties of equality.
m ABC = m CBD, then m CBD = m ABC.
6 0
3 years ago
Read 2 more answers
PLEASE HELP ASAP
Nat2105 [25]
A leg of a right triangle can never be greater then the hypotenuse

this formula will tell you if it's a valid right triangle
{?}^{2} + {?}^{2} = {?}^{2}
the first two question marks are the legs of the triangle while the one after the = is the hypotenuse.

THE STEPS:
{3}^{2} + { \sqrt{7}^{2} } = {4}^{2}
9 + {2.65}^{2} = 16
{2.65}^{2} = 7
9 + 7 = 16
ANSWER: \sqrt{7}
5 0
3 years ago
Other questions:
  • There are (53)2 ⋅ 50 hens in a bird enclosure. What is the total number of hens in the enclosure?
    13·2 answers
  • Solve 11x-6y=3; 5x-8y=-25
    5·1 answer
  • Khan academy plz help do not understand best answer will get brainliest
    7·1 answer
  • Ann's car can go 268 on 8 gallons of gas. during the weekend Ann used 3 gallons of gas.how far did she drive?.explain thanks and
    8·2 answers
  • Let m=x²-5. which equation is equivalent to (x²-5)²-3x²+15=-2 in term of m?
    5·1 answer
  • What number should be subtracted from -3/4 to get 5/6?
    6·1 answer
  • Please help me, i'll give you brainlyist <br> 20 points (maths)
    12·1 answer
  • You can view the question below. ​
    9·1 answer
  • Solve for x:<br> 4(1 – x) + 2 x = -3( x + 1)
    6·2 answers
  • 1. Sally has a bag of marbles. She has 3 different colored marbles in her bag – orange, black, and purple. If the probability of
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!