Help please i have little time

Please help me with the below question.

VMariaS [17]

By letting

$y = \displaystyle \sum_{n=0}^\infty c_n x^{n+r}$

we get derivatives

$y' = \displaystyle \sum_{n=0}^\infty (n+r) c_n x^{n+r-1}$

$y'' = \displaystyle \sum_{n=0}^\infty (n+r) (n+r-1) c_n x^{n+r-2}$

a) Substitute these into the differential equation. After a lot of simplification, the equation reduces to

$5r(r-1) c_0 x^{r-1} + \displaystyle \sum_{n=1}^\infty \bigg( (n+r+1) c_n + (n + r + 1) (5n + 5r + 1) c_{n+1} \bigg) x^{n+r} = 0$

Examine the lowest degree term $\left(x^{r-1}\right)$ , which gives rise to the indicial equation,

$5r (r - 1) + r = 0 \implies 5r^2 - 4r = r (5r - 4) = 0$

with roots at r = 0 and r = 4/5.

b) The recurrence for the coefficients $c_k$ is

$(k+r+1) c_k + (k + r + 1) (5k + 5r + 1) c_{k+1} = 0 \implies c_{k+1} = -\dfrac{c_k}{5k+5r+1}$

so that with r = 4/5, the coefficients are governed by

$c_{k+1} = -\dfrac{c_k}{5k+5} \implies \boxed{g(k) = -\dfrac1{5k+5}}$

c) Starting with $c_0=1$ , we find

$c_1 = -\dfrac{c_0}5 = -\dfrac15$

$c_2 = -\dfrac{c_1}{10} = \dfrac1{50}$

so that the first three terms of the solution are

$\displaystyle \sum_{n=0}^2 c_n x^{n + 4/5} = \boxed{x^{4/5} - \dfrac15 x^{9/5} + \frac1{50} x^{13/5}}$

4 0

2 years ago

What type of function is represented by the table of values below? 8 12 16 20

Dominik [7]

Answer:

To be precise, I think it's a pattern question, where you have to add 4 to each number. I'm not very sure though.

8 0

3 years ago

There are two spinners. The first spinner has three equal sectors labeled 1, 2 and 3. The second spinner has four equal sectors

sladkih [1.3K]

Given that the first spinner has three equal sectors labelled 1, 2 and 3; and the second spinner has equal sectors labelled 3, 4, 5 and 6.

The number of possible outcomes that do not show a 1 on the first spinner is 2 (i.e. the first spinner shows 2 or 3).

The number of possible outcomes that the second spinner show the number 4 is 1 (i.e. the second spinner shows 4)

In probability, the word 'and' goes with multiplication.

Therefore, the number of possible outcomes that do not show a 1 on the first spinner and show the number 4 on the second spinner is given by 2 x 1 = 2 possible outcomes.

i.e. the first spinner shows the number 2 and the second spinner shows the number 4 or the first spinner shows the number 3 and the second spinner shows the number 4.

5 0

3 years ago

Math help What are the solutions

Westkost [7]

Answer:

x² - 4x + 12

Step-by-step explanation:

x=2+2.82843i

x=2−2.82843i

6 0

2 years ago

At a corner gas station, the revenue R varies directly with the number g of gallons of gasoline sold. If the revenue is

vfiekz [6]

At a corner gas station, the revenue R varies directly with the number g of gallons of gasoline sold. If the revenue is $44.50 when the number of gallons sold is 10, find a linear equation that relates revenue R to the number g of gallons of gasoline. Then find the revenue R when the number of gallons of gasoline sold is 15.5.

Solution:
As the question mentioned the direct relationship between the quantities, hence
10 gallons of gasoline sold = $44.50
15.5 gallons of gasoline sold = $x
by cross multiplication, we get that
10x = 15.5 * 44.50
which implies that
x = 68.975
Thus by $68.975 revenue is obtained by selling 15.5 gallons of gasoline.

7 0

3 years ago