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
10 months 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]10 months 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
Raul is building a house. On day 1, he uses 3 bags of cement. On each of the following 3 days, he uses 3 times the number of bag
Ksivusya [100]
18bags 
Look if 1=3 2=9 3= 18  so he uses 18 bags!
8 0
3 years ago
I’m trying to get a lot of my geometry done. Please help.
Nataliya [291]

Answer:

do the work first. and u will get the answer easily.

6 0
3 years ago
What is the approximate value of a local maximum for the polynomial
statuscvo [17]

Answer:

D) 2.5

                                             Aaliyah- ツ

8 0
2 years ago
Evaluate the expression when x = 3/10 and y = - 2/15
Shkiper50 [21]

Answer:

6/15-3/10

Step-by-step explanation:

3x2/15=6/15

7 0
2 years ago
Read 2 more answers
PLEASE HELP ASAP!!!!!! Quadrilateral PQRS, with vertex P(-5, -3), undergoes a transformation to form quadrilateral P′Q′R′S′, wit
Nana76 [90]

Answer:

The vertex Q' is at (4,5)

Step-by-step explanation:

Given:

Quadrilateral PQRS undergoes a transformation to form a quadrilateral P'Q'R'S' such that the vertex point P(-5,-3) is transformed to P'(5,3).

Vertex point Q(-4,-5)

To find vertex Q'.

Solution:

Form the given transformation occuring the statement in standard form can be given as:

(x,y)\rightarrow (-x,-y)

The above transformation signifies the point reflection in the origin.

For the point P, the statement is:

P(-5,-3)\rightarrow P'(5,3)

So, for point Q, the transformation would be:

Q(-4,-5)\rightarrow Q'(-(-4),-(-5))

Since two negatives multiply to give a positive, so, we have:

Q(-4,-5)\rightarrow Q'(4,5)

5 0
2 years ago
Other questions:
  • Jimmy’s age is one year less than the sum of the ages of his siblings Serena and Tyler. Which equation represents Jimmy’s age?
    11·1 answer
  • Solve the following equation for x: 2x – 3y = 6.
    13·1 answer
  • 38 POINTS!! GEOMETRY HELP! Find the values of x and y. Prove your answers.
    13·1 answer
  • Use integer rules to evaluate: 85 + (-96) *
    15·1 answer
  • Divide. 8.844÷0.04 enter the anwser in the box
    13·1 answer
  • 3(2x - 5) - 8x = -2(x - 1)
    8·1 answer
  • WILL GIVE BRAINLIEST to whoever solves this entire problem (completes all the steps). I WILL REPORT if you just steal the points
    12·1 answer
  • 44%
    8·1 answer
  • What is greater than in mathematice
    10·1 answer
  • What is the slope of a line perpendicular to the line whose equation is 3x + 6y = –72. Fully reduced
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!