All categories

2 years ago

Help please????????????

Mathematics

1 answer:

fredd [130]2 years ago

7 0

Answer:

hiiii i total can help im doing this in school

Step-by-step explanation:

You might be interested in

Find two power series solutions of the given differential equation about the ordinary point x = 0. compare the series solutions

monitta

I don't know what method is referred to in "section 4.3", but I'll suppose it's reduction of order and use that to find the exact solution. Take $z=y'$ , so that $z'=y''$ and we're left with the ODE linear in $z$ :

$y''-y'=0\implies z'-z=0\implies z=C_1e^x\implies y=C_1e^x+C_2$

Now suppose $y$ has a power series expansion

$y=\displaystyle\sum_{n\ge0}a_nx^n$
$\implies y'=\displaystyle\sum_{n\ge1}na_nx^{n-1}$
$\implies y''=\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}$

Then the ODE can be written as

$\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}-\sum_{n\ge1}na_nx^{n-1}=0$

$\displaystyle\sum_{n\ge2}n(n-1)a_nx^{n-2}-\sum_{n\ge2}(n-1)a_{n-1}x^{n-2}=0$

$\displaystyle\sum_{n\ge2}\bigg[n(n-1)a_n-(n-1)a_{n-1}\bigg]x^{n-2}=0$

All the coefficients of the series vanish, and setting $x=0$ in the power series forms for $y$ and $y'$ tell us that $y(0)=a_0$ and $y'(0)=a_1$ , so we get the recurrence

$\begin{cases}a_0=a_0\\\\a_1=a_1\\\\a_n=\dfrac{a_{n-1}}n&\text{for }n\ge2\end{cases}$

We can solve explicitly for $a_n$ quite easily:

$a_n=\dfrac{a_{n-1}}n\implies a_{n-1}=\dfrac{a_{n-2}}{n-1}\implies a_n=\dfrac{a_{n-2}}{n(n-1)}$

and so on. Continuing in this way we end up with

$a_n=\dfrac{a_1}{n!}$

so that the solution to the ODE is

$y(x)=\displaystyle\sum_{n\ge0}\dfrac{a_1}{n!}x^n=a_1+a_1x+\dfrac{a_1}2x^2+\cdots=a_1e^x$

We also require the solution to satisfy $y(0)=a_0$ , which we can do easily by adding and subtracting a constant as needed:

$y(x)=a_0-a_1+a_1+\displaystyle\sum_{n\ge1}\dfrac{a_1}{n!}x^n=\underbrace{a_0-a_1}_{C_2}+\underbrace{a_1}_{C_1}\displaystyle\sum_{n\ge0}\frac{x^n}{n!}$

4 0

2 years ago

A football coach gives half his players a protein bar to eat before each game; the other players do not receive the protein bar.

timofeeve [1]

To make the sure the results did not happen by chance, the coach can ensure that the protein bars are the only difference between the two groups. For example, he can make sure that both groups are drinking the same amounts of water and getting in the same amount of practice; that way, he will be able to determine that one group is performing better due to the protein bars only. Also, the coach can repeat this experiment multiple times to make sure that the results weren't a random, one-time thing.

4 0

3 years ago

Simplify 9 to the 2nd over 9 to the 7th

Allisa [31]

3387⁄200000000 or 0.00001693508

5 0

2 years ago

The SAT mathematics scores in the state of Florida for this year are approximately normally distributed with a mean of 500 and a

Sergeeva-Olga [200]

Because it will be half2 standard deviation which will include 500-700.
2std will include from 300-700.
so it will be half of 95%.

7 0

3 years ago

Please help! I’ll give brainliest!!

inn [45]

Answer:

1.a part, share, or number considered in comparative relation to a whole.

1/3 = 2/6

2.If a proposition is true, then we say it has a truth value of "true"; if a proposition is false, its truth value is "false". For example, "Grass is green", and "2 + 5 = 5" are propositions. The first proposition has the truth value of "true" and the second "false".

Don't confuse cross multiplication with cross cancellation. You use cross cancellation to make multiplying fractions easier. Set the two cross products equal to each other. 1 · 2 = 2 and 0.5 · f = 0.5f.

5 0

3 years ago