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
Alex
1 year ago
6

Find two power series solutions of the given differential equation about the ordinary point x = 0. y'' + xy = 0

Mathematics
1 answer:
nalin [4]1 year ago
6 0

Answer:

First we write y and its derivatives as power series:

y=∑n=0∞anxn⟹y′=∑n=1∞nanxn−1⟹y′′=∑n=2∞n(n−1)anxn−2

Next, plug into differential equation:

(x+2)y′′+xy′−y=0

(x+2)∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0

x∑n=2∞n(n−1)anxn−2+2∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0

Move constants inside of summations:

∑n=2∞x⋅n(n−1)anxn−2+∑n=2∞2⋅n(n−1)anxn−2+∑n=1∞x⋅nanxn−1−∑n=0∞anxn=0

∑n=2∞n(n−1)anxn−1+∑n=2∞2n(n−1)anxn−2+∑n=1∞nanxn−∑n=0∞anxn=0

Change limits so that the exponents for  x  are the same in each summation:

∑n=1∞(n+1)nan+1xn+∑n=0∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−∑n=0∞anxn=0

Pull out any terms from sums, so that each sum starts at same lower limit  (n=1)

∑n=1∞(n+1)nan+1xn+4a2+∑n=1∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−a0−∑n=1∞anxn=0

Combine all sums into a single sum:

4a2−a0+∑n=1∞(2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an)xn=0

Now we must set each coefficient, including constant term  =0 :

4a2−a0=0⟹4a2=a0

2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an=0

We would usually let  a0  and  a1  be arbitrary constants. Then all other constants can be expressed in terms of these two constants, giving us two linearly independent solutions. However, since  a0=4a2 , I’ll choose  a1  and  a2  as the two arbitrary constants. We can still express all other constants in terms of  a1  and/or  a2 .

an+2=−(n+1)nan+1+(n−1)an2(n+2)(n+1)

a3=−(2⋅1)a2+0a12(3⋅2)=−16a2=−13!a2

a4=−(3⋅2)a3+1a22(4⋅3)=0=04!a2

a5=−(4⋅3)a4+2a32(5⋅4)=15!a2

a6=−(5⋅4)a5+3a42(6⋅5)=−26!a2

We see a pattern emerging here:

an=(−1)(n+1)n−4n!a2

This can be proven by mathematical induction. In fact, this is true for all  n≥0 , except for  n=1 , since  a1  is an arbitrary constant independent of  a0  (and therefore independent of  a2 ).

Plugging back into original power series for  y , we get:

y=a0+a1x+a2x2+a3x3+a4x4+a5x5+⋯

y=4a2+a1x+a2x2−13!a2x3+04!a2x4+15!a2x5−⋯

y=a1x+a2(4+x2−13!x3+04!x4+15!x5−⋯)

Notice that the expression following constant  a2  is  =4+  a power series (starting at  n=2 ). However, if we had the appropriate  x -term, we would have a power series starting at  n=0 . Since the other independent solution is simply  y1=x,  then we can let  a1=c1−3c2,   a2=c2 , and we get:

y=(c1−3c2)x+c2(4+x2−13!x3+04!x4+15!x5−⋯)

y=c1x+c2(4−3x+x2−13!x3+04!x4+15!x5−⋯)

y=c1x+c2(−0−40!+0−31!x−2−42!x2+3−43!x3−4−44!x4+5−45!x5−⋯)

y=c1x+c2∑n=0∞(−1)n+1n−4n!xn

Learn more about constants here:

brainly.com/question/11443401

#SPJ4

You might be interested in
20 less than a number
agasfer [191]

Answer:

20 less than a number is equal to the number.

Step-by-step explanation:

8 0
3 years ago
Drag each measure to a box on the right to match the measure on the left. 5 feet10,560 feet48 inches126 inches 2 miles 123 yards
charle [14.2K]

Answer:

4 feets = 48 inches

2 miles = 10,560 feets

1 2/3 yards = 5 feets

3 1/2 yards = 126 inches

Step-by-step explanation:

Recall :

1 Feet = 12 inches

1 MILE = 5280 feets

1 yard = 36 inches

1 yard = 3 feets

If 1 feet = 12 inches

4 feets will be ; (12 * 4) inches = 48 inches

If 1 mile = 5280 feets

2 miles will be ; (5280 * 2) = 10,560 feets

If 1 yard = 3 feets

1 2/3 yards will be ; (5 /3) * 3 = 5 feets

If 1 yard = 36 inches

3 1/2 yards will be ; (7 /2 ) * 36 = 126 inches

8 0
3 years ago
Find the values of the variables.
Jlenok [28]

Answer:

The fourth option - t = 3, y = 5

Step-by-step explanation:

-8 + t becomes -5, thus t is 3 since -8 + 3 = -5.

-12 becomes -2y - 2, so we can set these equal to each other as -2y - 2 = -12 and solve from there. Add 2 to both sides, -2y = -10, then divide by -2 on both sides, y = 5.

6 0
3 years ago
Read 2 more answers
Brand A scooter has a top speed that goes 2 miles per hour faster than Brand B. if after hours, Brand A scooter traveled 24 mile
Leni [432]
Your answer would be 22mph
3 0
3 years ago
Read 2 more answers
Last year, of business owners gave a holiday gift to their employees. A survey of business owners conducted this year indicates
IRINA_888 [86]
It would be 4 business owners will gift their employees
4 0
2 years ago
Other questions:
  • Which values are solutions to the inequality below?<br> Check all that apply
    15·1 answer
  • Let f be the function given by f(x)=ln x/x for all x 0. the derivative of f is given by
    14·1 answer
  • Please answer !!! To solve the system of equations below, Luanne isolated the variable y in the first equation and then substitu
    14·2 answers
  • What is the answer to 8-x=3x
    15·2 answers
  • Let f(x) = x + 8 and g(x) = x2 − 6x − 7. Find f(g(2)). (1 point)<br><br><br> −7<br> −3<br> 10<br> 33
    10·1 answer
  • 2)<br> -<br> 2-1 and-7v&gt;14<br> 8<br> -12<br> -10<br> -8<br> -4<br> -2<br> 0
    12·1 answer
  • Noralie has a home-based business putting on children’s parties. She charges $45 to design the party and then $5.50 per child. W
    6·1 answer
  • Can you help me with this please. <br>4)<br>5)<br>6)<br>7)<br>8)<br>9)<br>10)​
    12·1 answer
  • -4b + 8c + 12 - 8b - 2c + 6. what is the value of the expression when b=2 and c= -3? Enter your answer.
    11·1 answer
  • Plz simplify with steps
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!