1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
liubo4ka [24]
3 years ago
9

X^2-6=2-18x solve by completing the square

Mathematics
1 answer:
AnnyKZ [126]3 years ago
6 0
X^1=-9[89],x^2=-9+[89]
You might be interested in
Xy''+2y'-xy by frobenius method
aalyn [17]
First note that x=0 is a regular singular point; in particular x=0 is a pole of order 1 for \dfrac2x.

We seek a solution of the form

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

where r is to be determined. Differentiating, we have

y'=\displaystyle\sum_{n\ge0}(n+r)a_nx^{n+r-1}
y''=\displaystyle\sum_{n\ge0}(n+r)(n+r-1)a_nx^{n+r-2}

and substituting into the ODE gives

\displaystyle x\sum_{n\ge0}(n+r)(n+r-1)a_nx^{n+r-2}+2\sum_{n\ge0}(n+r)a_nx^{n+r-1}-x\sum_{n\ge0}a_nx^{n+r}=0
\displaystyle \sum_{n\ge0}(n+r)(n+r-1)a_nx^{n+r-1}+2\sum_{n\ge0}(n+r)a_nx^{n+r-1}-\sum_{n\ge0}a_nx^{n+r+1}=0
\displaystyle \sum_{n\ge0}(n+r)(n+r+1)a_nx^{n+r-1}-\sum_{n\ge0}a_nx^{n+r+1}=0
\displaystyle r(r+1)a_0x^{r-1}+(r+1)(r+2)a_1x^r+\sum_{n\ge2}(n+r)(n+r+1)a_nx^{n+r-1}-\sum_{n\ge0}a_nx^{n+r+1}=0
\displaystyle r(r+1)a_0x^{r-1}+(r+1)(r+2)a_1x^r+\sum_{n\ge2}(n+r)(n+r+1)a_nx^{n+r-1}-\sum_{n\ge2}a_{n-2}x^{n+r-1}=0
\displaystyle r(r+1)a_0x^{r-1}+(r+1)(r+2)a_1x^r+\sum_{n\ge2}\bigg((n+r)(n+r+1)a_n-a_{n-2}\bigg)x^{n+r-1}=0

The indicial polynomial, r(r+1), has roots at r=0 and r=-1. Because these roots are separated by an integer, we have to be a bit more careful, but we'll get back to this later.

When r=0, we have the recurrence

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

valid for n\ge2. When n=2k, with k\in\{0,1,2,3,\ldots\}, we find

a_0=a_0
a_2=\dfrac{a_0}{3\cdot2}=\dfrac{a_0}{3!}
a_4=\dfrac{a_2}{5\cdot4}=\dfrac{a_0}{5!}
a_6=\dfrac{a_4}{7\cdot6}=\dfrac{a_0}{7!}

and so on, with a general pattern of

a_{n=2k}=\dfrac{a_0}{(2k+1)!}

Similarly, when n=2k+1 for k\in\{0,1,2,3,\ldots\}, we find

a_1=a_1
a_3=\dfrac{a_1}{4\cdot3}=\dfrac{2a_1}{4!}
a_5=\dfrac{a_3}{6\cdot5}=\dfrac{2a_1}{6!}
a_7=\dfrac{a_5}{8\cdot7}=\dfrac{2a_1}{8!}

and so on, with the general pattern

a_{n=2k+1}=\dfrac{2a_1}{(2k+2)!}

So the first indicial root admits the solution

y=\displaystyle a_0\sum_{k\ge0}\frac{x^{2k}}{(2k+1)!}+a_1\sum_{k\ge0}\frac{x^{2k+1}}{(2k+2)!}
y=\displaystyle \frac{a_0}x\sum_{k\ge0}\frac{x^{2k+1}}{(2k+1)!}+\frac{a_1}x\sum_{k\ge0}\frac{x^{2k+2}}{(2k+2)!}
y=\displaystyle \frac{a_0}x\sum_{k\ge0}\frac{x^{2k+1}}{(2k+1)!}+\frac{a_1}x\sum_{k\ge0}\frac{x^{2k+2}}{(2k+2)!}

which you can recognize as the power series for \dfrac{\sinh x}x and \dfrac{\cosh x}x.

To be more precise, the second series actually converges to \dfrac{\cosh x-1}x, which doesn't satisfy the ODE. However, remember that the indicial equation had two roots that differed by a constant. When r=-1, we may seek a second solution of the form

y=cy_1\ln x+x^{-1}\displaystyle\sum_{n\ge0}b_nx^n

where y_1=\dfrac{\sinh x+\cosh x-1}x. Substituting this into the ODE, you'll find that c=0, and so we're left with

y=x^{-1}\displaystyle\sum_{n\ge0}b_nx^n
y=\dfrac{b_0}x+b_1+b_2x+b_3x^2+\cdots

Expanding y_1, you'll see that all the terms x^n with n\ge0 in the expansion of this new solutions are already accounted for, so this new solution really only adds one fundamental solution of the form y_2=\dfrac1x. Adding this to y_1, we end up with just \dfrac{\sinh x+\cosh x}x.

This means the general solution for the ODE is

y=C_1\dfrac{\sinh x}x+C_2\dfrac{\cosh x}x
3 0
3 years ago
Write an expression for "the quotient of 9 and c."
frez [133]
Quotient is when something is divided by something. therefore, an equation could be 9/c
3 0
3 years ago
 is the diameter of a circle with center P. The coordinates of the endpoints of the diameter are (1, 3) and (6, 6). What are th
Oksana_A [137]
See picture for answer.

3 0
3 years ago
Read 2 more answers
What is 17. 60 divided by 8 in short division?
lesantik [10]
The answer's 2.2
I'm sorry I can't show you, but LaTex doesn't allow you to show short division.
You can find out how to do it here, on Wikipedia's page. https://www.wikiwand.com/en/Short_division
8 0
3 years ago
Translate the sentence into an inequality. <br> Six subtracted from b is at most 21.
o-na [289]
This is my answer that's a less than or equal sign

4 0
3 years ago
Other questions:
  • A package of 12 cans of beans cost $6.84. what is the cost of ONE cost of beans?
    7·2 answers
  • Dean’s car weighs 1 1/4 tons. How many pounds does this car away?
    15·2 answers
  • Solve for x -9 = -7
    8·2 answers
  • Plz help ASAP PLZZZ!!!
    6·2 answers
  • How to find x for this equation : 3x-3(x-1)=3
    8·1 answer
  • Victor makes decorations for his birthday party. He needs 5/6 yard of ribbon for each decoration. How much ribbon does Victor ne
    9·1 answer
  • 2/12= blank 48 gffnvbnvbnvbfffnvbnbvffffffffbvcnffffffffffnbvnvbnbvnbvffffffffffffffffff
    11·1 answer
  • Please make sure you do it right.
    14·2 answers
  • 4(w + 1) = -6; w = <br> help asap
    11·2 answers
  • For one binomial experiment, n1 = 75 binomial trials produced r1 = 30 successes. For a second independent binomial experiment, n
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!