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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\}}

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escribe la ecuación canónica de la elipse con centro en el origen que pasa por el punto (2,1) y cuyo eje menor mide 4 y se encue
mamaluj [8]

Answer:

Ur mummm u cheater

Step-by-step explanation:

6 0
3 years ago
What much expression represents a rational number?
Pie

Answer:

\displaystyle \frac{2}{7}+\sqrt{121}

Step-by-step explanation:

<u>Rational Numbers</u>

A rational number is any number that can be expressed as a fraction

\displaystyle \frac{a}{b}, \ b\neq 0

for a and b any integer and b different from 0.

As a consequence, any number that cannot be expressed as a fraction or rational number is defined as an Irrational number.

Let's analyze each one of the given options

\displaystyle \frac{5}{9}+\sqrt{18}

The first part of the number is indeed a rational number, but the second part is a square root whose result cannot be expressed as a rational, thus the number is not rational

\pi + \sqrt{16}

The second part is an exact square root (resulting 4) but the first part is a known irrational number called pi. It's not possible to express pi as a fraction, thus the number is irrational

\displaystyle \frac{2}{7}+\sqrt{121}

The square root of 121 is 11. It makes the whole number a sum of a rational number plus an integer, thus the given number is rational

\displaystyle \frac{3}{10}+\sqrt{11}

As with the first number, the square root is not exact. The sum of a rational number plus an irrational number gives an irrational number.

Correct option:

\boxed{\displaystyle \frac{2}{7}+\sqrt{121}}

6 0
3 years ago
PLEASE HELP ME !!!!!!
Ad libitum [116K]

Step-by-step explanation:

In this question as two straight lines intersect each other

we use vertically opposite angle axiom

so 74 degree = 2x.

then x= 74/2= 37degree

6 0
2 years ago
Read 2 more answers
13. Given the volume, V, of the rectangular prism below, find h, the height of the prism. V = (x ^ 2 + 2x)/(x + 1) l= 2x + 4 w=
NikAS [45]

Given:

Volume of a rectangular prism is:

V=\dfrac{x^2+2x}{x+1}

Dimensions of the rectangular prism are:

l=2x+4

w=\dfrac{x}{8}

To find:

The height of the rectangular prism.

Solution:

The volume of a rectangular prism is:

V=l\times w\times h

After substituting the values, we get

\dfrac{x^2+2x}{x+1}=(2x+4)\times \dfrac{x}{8}\times h

\dfrac{x(x+2)}{x+1}=\dfrac{2x(x+2)}{8}\times h

\dfrac{x(x+2)}{x+1}\times \dfrac{8}{2x(x+2)}=h

\dfrac{1}{x+1}\times \dfrac{8}{2}=h

\dfrac{4}{x+1}=h

Therefore, the correct option is C.

8 0
3 years ago
Question 19 of 40
Anna71 [15]

The correct definition of an angle is D: A shape formed by two intersecting lines from a common point.

<h3>What is a line segment?</h3>

A line segment is extended infinitely in both directions whereas a 'ray' is a line segment that has one endpoint and extends infinitely in the other direction.

Now, an 'angle' is formed by two rays having a common end-point.

As the angle has a common endpoint,

Therefore it is not possible to form an angle by intersecting two rays having different endpoints.

Hence, option D is correct.

Learn more about line;

brainly.com/question/3455295

#SPJ1

3 0
2 years ago
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