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Alex787 [66]
3 years ago
6

he yearbook club had a meeting. The meeting had 24 people, which is three-fourths of the club. How many people are in the club

Mathematics
1 answer:
WINSTONCH [101]3 years ago
4 0

Answer:

32 people

Step-by-step explanation:

If 3/4 equals 24, we need to find the other fourth. We can divided the total number of people by the numerator. 24/8, then add 8, which equals 32.

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Read 2 more answers
For the differential equation 3x^2y''+2xy'+x^2y=0 show that the point x = 0 is a regular singular point (either by using the lim
Svetlanka [38]
Given an ODE of the form

y''(x)+p(x)y'(x)+q(x)y(x)=f(x)

a regular singular point x=c is one such that p(x) or q(x) diverge as x\to c, but the limits of (x-c)p(x) and (x-c)^2q(x) as x\to c exist.

We have for x\neq0,

3x^2y''+2xy'+x^2y=0\implies y''+\dfrac2{3x}y'+\dfrac13y=0

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We then look for a series solution about the regular singular point x=0 of the form

y=\displaystyle\sum_{n\ge0}a_nx^{n+k}

Substituting into the ODE gives

\displaystyle3x^2\sum_{n\ge0}a_n(n+k)(n+k-1)x^{n+k-2}+2x\sum_{n\ge0}a_n(n+k)x^{n+k-1}+x^2\sum_{n\ge0}a_nx^{n+k}=0

\displaystyle3\sum_{n\ge2}a_n(n+k)(n+k-1)x^{n+k}+3a_1k(k+1)x^{k+1}+3a_0k(k-1)x^k
\displaystyle+2\sum_{n\ge2}a_n(n+k)x^{n+k}+2a_1(k+1)x^{k+1}+2a_0kx^k
\displaystyle+\sum_{n\ge2}a_{n-2}x^{n+k}=0

From this we find the indicial equation to be

(3(k-1)+2)ka_0=0\implies k=0,\,k=\dfrac13

Taking k=\dfrac13, and in the x^{k+1} term above we find a_1=0. So we have

\begin{cases}a_0=1\\a_1=0\\\\a_n=-\dfrac{a_{n-2}}{n(3n+1)}&\text{for }n\ge2\end{cases}

Since a_1=0, all coefficients with an odd index will also vanish.

So the first three terms of the series expansion of this solution are

\displaystyle\sum_{n\ge0}a_nx^{n+1/3}=a_0x^{1/3}+a_2x^{7/3}+a_4x^{13/3}

with a_0=1, a_2=-\dfrac1{14}, and a_4=\dfrac1{728}.
6 0
3 years ago
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