MATHEMATICS COLLEGE

Solve this differential Equation by using power series
y''-x^2y=o

Answers

Answer 1
Answer:

We're looking for a solution

y=\displaystyle\sum_(n=0)^\infty a_nx^n

which has second derivative

y''=\displaystyle\sum_(n=2)^\infty n(n-1)a_nx^(n-2)=\sum_(n=0)^\infty(n+2)(n+1)a_(n+2)x^n

Substituting these into the ODE gives

\displaystyle\sum_(n=0)^\infty(n+2)(n+1)a_(n+2)x^n-\sum_(n=0)^\infty a_nx^(n+2)=0

\displaystyle\sum_(n=0)^\infty(n+2)(n+1)a_(n+2)x^n-\sum_(n=2)^\infty a_(n-2)x^n=0

\displaystyle2a_2+6a_3x+\sum_(n=2)^\infty(n+2)(n+1)a_(n+2)x^n-\sum_(n=2)^\infty a_(n-2)x^n=0

\displaystyle2a_2+6a_3x+\sum_(n=2)^\infty\bigg((n+2)(n+1)a_(n+2)-a_(n-2)\bigg)x^n=0

Right away we see a_2=a_3=0, and the coefficients are given according to the recurrence

\begin{cases}a_0=y(0)\na_1=y'(0)\na_2=0\na_3=0\nn(n-1)a_n=a_(n-4)&\text{for }n\ge4\end{cases}

There's a dependency between terms in the sequence that are 4 indices apart, so we consider 4 different cases.

  • If n=4k, where k\ge0 is an integer, then

k=0\implies n=0\implies a_0=a_0

k=1\implies n=4\implies a_4=(a_0)/(4\cdot3)=\frac2{4!}a_0

k=2\implies n=8\implies a_8=(a_4)/(8\cdot7)=(6\cdot5\cdot2)/(8!)a_0

k=3\implies n=12\implies a_(12)=(a_8)/(12\cdot11)=(10\cdot9\cdot6\cdot5\cdot2)/(12!)a_0

and so on, with the general pattern

a_(4k)=(a_0)/((4k)!)\displaystyle\prod_(i=1)^k(4i-2)(4i-3)

  • If n=4k+1, then

k=0\implies n=1\implies a_1=a_1

k=1\implies n=5\implies a_5=(a_1)/(5\cdot4)=(3\cdot2)/(5!)a_1

k=2\implies n=9\implies a_9=(a_5)/(9\cdot8)=(7\cdot6\cdot3\cdot2)/(9!)a_1

k=3\implies n=13\implies a_(13)=(a_9)/(13\cdot12)=(11\cdot10\cdot7\cdot6\cdot3\cdot2)/(13!)a_1

and so on, with

a_(4k+1)=(a_1)/((4k+1)!)\displaystyle\prod_(i=1)^k(4i-1)(4i-2)

  • If n=4k+2 or n=4k+3, then

a_2=0\implies a_6=a_(10)=\cdots=a_(4k+2)=0

a_3=0\implies a_7=a_(11)=\cdots=a_(4k+3)=0

Then the solution to this ODE is

\boxed{y(x)=\displaystyle\sum_(k=0)^\infty a_(4k)x^(4k)+\sum_(k=0)^\infty a_(4k+1)x^(4k+1)}

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Kenny worked 6.5 hours on Monday, and 5 hours on Tuesday. If he received a check for $143.75 for working those 2 days, how much does Kenny make per hour?

Kenny bought 20 packs of baseball cards. There are 12 cards in each pack. How many cards did Kenny buy?

Answers

Answer:

240

Step-by-step explanation:

20 x 12 = 240
Kenny bought 240 baseball cards.

The expression 3a + 2b + c / d gives the average number of points a team earns in d games by winning a games in regulation, b games in overtime and losing c games in overtime. What is the average number of points earned for a team that wins 10 games in regulation, wins 8 in overtime, loses 6 in overtime, and loses 4 in regulation?

Answers

Answer:

8

Step-by-step explanation:

10+8-6-4=8

Given the regular polygon, find the measure of each numbered angle.2

Answers

Weare given a regular figure, and asked for the measure of some indicated angles as shown below:

Notice that this is regolar figure (all sides of the same size) and also that is has 8 sides. Therefore, the central angle (360 degrees) must be divided by 8 to find the measure of angle <1:

angle <1 = 360 / 8 = 45 degrees.

Now, since the little triangles that the polygon is divided into, are all isosceles triangles (have the two sides connected to the center of the polygon, equal, then the other two angles inside each triangle must be equal (let's call them "x").

Now, for each little triangle we use the property that the addition of its internal angles must equal 180 degrees:

<1 + x + x = 180

45 + 2 x = 180

2 x = 180 - 45

2 x = 135

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Then <1 = 45 degrees , and <2 = 67.5 degrees.

There are two boxes of Lego pieces. One box has 200 pieces and the other box has 115 pieces. Which of the following best describes how 4 people will divide the Lego pieces as evenly as possible?three people will get 77 pieces
One person will get 76 pieces



Three people will get 78 pieces
One person will get 77 pieces



Three people will get 79 pieces
One person will get 78 pieces



Three people will get 76 pieces
One person will get 75 pieces

Answers

The answer is C: three people will get 79 pieces and one person will get 78 pieces. 
79x3=237
237+78=315
There were 315 total pieces (200+115) so it has to be C. 
Hope this helps!
three people will get 79 pieces  
one person will get 78 pieces
200+115=315
79x3=237+78=315

Rita earns $18.00 per hr. If she gets a 6% raise, what will be her new hourly wage?

Answers

Answer:

19.06

Step-by-step explanation:

Take her wage and multiply by the percent increase

18*.06 =1.08

Add this to her original wage

18+1.06

19.06 is the new wage

Any help will mark brainliest

Answers

4 x 2 tens = 80, 28 x 14 = 392
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