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
Illusion [34]
3 years ago
7

1) Use power series to find the series solution to the differential equation y'+2y = 0 PLEASE SHOW ALL YOUR WORK, OR RISK LOSING

ALL POINTS!!
Mathematics
1 answer:
iogann1982 [59]3 years ago
7 0

If

y=\displaystyle\sum_{n=0}^\infty a_nx^n

then

y'=\displaystyle\sum_{n=1}^\infty na_nx^{n-1}=\sum_{n=0}^\infty(n+1)a_{n+1}x^n

The ODE in terms of these series is

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

\displaystyle\sum_{n=0}^\infty\bigg(a_{n+1}+2a_n\bigg)x^n=0

\implies\begin{cases}a_0=y(0)\\(n+1)a_{n+1}=-2a_n&\text{for }n\ge0\end{cases}

We can solve the recurrence exactly by substitution:

a_{n+1}=-\dfrac2{n+1}a_n=\dfrac{2^2}{(n+1)n}a_{n-1}=-\dfrac{2^3}{(n+1)n(n-1)}a_{n-2}=\cdots=\dfrac{(-2)^{n+1}}{(n+1)!}a_0

\implies a_n=\dfrac{(-2)^n}{n!}a_0

So the ODE has solution

y(x)=\displaystyle a_0\sum_{n=0}^\infty\frac{(-2x)^n}{n!}

which you may recognize as the power series of the exponential function. Then

\boxed{y(x)=a_0e^{-2x}}

You might be interested in
Which ordered pair would form a proportional relationship with the point graphed below
matrenka [14]
D (80,-30) so on so far
8 0
3 years ago
Read 2 more answers
The diameter of the circle below is 42 cm. Work out the radius of the circle.​
creativ13 [48]

Answer:

r = 21 cm

Step-by-step explanation:

The radius of a circle is 1/2 of the diameter

r = 1/2d

r = 1/2(42)

r = 21 cm

8 0
2 years ago
I wanted to walk one mile each day in the morning I walked 4/10 if a mile and in the evening I walked 2/5 of a mile what was the
Vitek1552 [10]
4/10 is also equal to 2/5 and 2+2 equals four therefore you would get 4/5 as your final answer
7 0
3 years ago
Read 2 more answers
Cloverdale Middle School is buying new helmets for the football team. Each helmet costs $89.95. How much money does the school s
seraphim [82]

Answer:

4047.75$

Step-by-step explanation:

4 0
2 years ago
From 248 to 200 workers
Yuri [45]
Are you asking what is 248-200? If so that would be 48. 200-248=248
3 0
3 years ago
Read 2 more answers
Other questions:
  • cameron is 3 years old older than karlee. karlee is two years older than denise. the sum of cameron,karlee,and denise age is 49.
    15·1 answer
  • The shampoo you like to use is on sale. But you only get the lower price if you buy 3 bottles. You want to know how much you'll
    14·2 answers
  • (6 2/5 + 1 4/5) + 3 1/5
    12·2 answers
  • Riding a proportion of 15 is 1 and a half percent of what number​
    11·1 answer
  • Ayudeme a saber cuales son las fracciones de 2/8 3/5 y 3/2<br> por favor
    12·1 answer
  • If angle 1 and angle 7 are vertical angles and angle 1 and angle 2 are supplementary adjacent and the measure of angle 2 is 123
    15·1 answer
  • X = -5<br> y = -6<br><br> What is the solution set?<br><br> {(-6, -5)}<br> {(-5, -6)}<br> Ø
    15·1 answer
  • The outline of Sara's family room, including the entry, is shown.
    9·1 answer
  • The sum of 10x + 6 and -x + 8
    9·1 answer
  • Statistics 4. Hol (a) The average age of 5 students is 9 years. Out of them, the ages of 4 students are 5, 7, 8, and 15 years. W
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!