Mathematics, 29.11.2019 06:31 niyyyareligion

Let be a real constant. consider the equation with boundary conditions and . for certain discrete values of , this equation can have non-zero solutions. find the three smallest values of for which this is the case.

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Answer from: 24lbriscoe

This linear ODE has characteristic equation

r^2-4r+a=0

with roots $r=2\pm\sqrt{4-a}$ , which gives solutions of the form

$y_c=C_1e^{(2+\sqrt{4-a})x}+C_2e^{(2-\sqrt{4-a})x}$

There are three cases to consider:

(1) If

, then the solution will be exactly what we see above. However, the initial conditions force both C_1=C_2=0

.

(2) If

, we're left with

$y_c=C_1e^{2x}+C_2xe^{2x}$

where $xe^{2x}$ is added to the solution set to account for a second solution that is linearly independent of the first solution. Again, we get C_1=C_2=0

.

(3) If

, then the square root introduces a factor of

that admits the solution

$y_c=C_1e^{2x}\cos(\sqrt{4-a}x)+C_2e^{2x}\sin(\sqrt{4-a}x)$

In this case, we arrive at C_1=0

, and from the second condition we get

$0=C_2e^{16}\sin(8\sqrt{4-a})$

In order that $C_2\neq0$ , we require that $8\sqrt{4-a}=n\pi$ , where

is any integer. Solving for

, we get

$8\sqrt{4-a}=n\pi\implies a=\dfrac{256-n^2\pi^2}{64}$

When n=0

, we arrive at a=4

, but remember that we're assuming that

, so logically the three smallest values of

that are allowed occur for n=1,2,3

. (

, so we can just look at positive integers

.)

Unfortunately, I'm not sure exactly what's going on next. Checking with a computer, the solution is supposed to be

$a=4+4n^2\pi^2$

(Again, not sure why this is the case, but let's move on.) When n=1,2,3

, we have the least values, which are, respectively,

$a=4+4\pi^2$
$a=4+16\pi^2$
$a=4+36\pi^2$

Answer from: Quest

answer

the correct answer is

$x=-5,x=7$

explanation

the given function is

$\frac{ {x}^{2} - 3x - 28 }{x {}^{2} - 2x - 35}$

the excluded values are obtained by solving:

${x}^{2} - 2x - 35 = 0$

we factor to obtain:

$(x + 5)(x - 7)= 0$

the excluded values are

$x=-5,x=7$

What are the excluded values of x for.. (look at picture)

Answer from: Quest

solution:

we have to construct

∠m nt ≅ ∠p qr

the steps to draw this construction are

step 1: use a compass to draw an arc from point q which intersects the side pq at point a and the side qr at point b.

step 2: draw a segment nt and use the same width of the compass to draw an arc from point n which intersects the segment nt at a point x.

step 3:

step 4: join points n and y using a straightedge.

step 3:

option (d).maintaining the same width of the compass as ab, draw an arc from point x such that it intersects the arc drawn from n in a point y.

to draw congruent angles , 1. width of compasses while drawing two angles should be same i.e from point at which angle has to be drawn, 2.the distance between the two points where the compass intersects the two sides should be same while drawing any two congruent angles.

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Let be a real constant. consider the equation with boundary conditions and . for certain discrete...

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