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
Mashutka [201]
3 years ago
6

. The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to nd

a second solution y2(x) of the homogeneous equation. x 2 y 00 7xy0 + 16y = 0; y1 = x 4
Mathematics
1 answer:
Blababa [14]3 years ago
6 0

The question is:

The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation

x²y'' - 7xy' + 16y = 0; y1 = x^4

Answer:

The second solution y2 is

A(x^4)lnx

Step-by-step explanation:

Given the homogeneous differential equation

x²y'' - 7xy' + 16y = 0

And a solution: y1 = x^4

We need to find a second solution y2 using the method of reduction of order.

Let y2 = uy1

=> y2 = ux^4

Since y2 is also a solution to the differential equation, it also satisfies it.

Differentiate y2 twice in succession with respect to x, to obtain y2' and y2'' and substitute the resulting values into the original differential equation.

y2' = u'. x^4 + u. 4x³

y2'' = u''. x^4 + u'. 4x³ + u'. 4x³ + u. 12x²

= u''. x^4 + u'. 8x³ + u. 12x²

Now, using these values in the original equation,

x²(u''. x^4 + u'. 8x³ + u. 12x²) - 7x(u'. x^4 + u. 4x³)+ 16(ux^4) = 0

x^6u'' + 8x^5u' + 12x^4u - 7x^5u' - 28x^4u + 16x^4u = 0

x^6u'' + x^5u' = 0

xu'' = -u'

Let w = u'

Then w' = u''

So

xw' = -w

w'/w = -1/x

Integrating both sides

lnw = -lnx + C

w = Ae^(-lnx) (where A = e^C)

w = A/x

But w = u'

So,

u' = A/x

Integrating this

u = Alnx

Since

y2 = uy1

We have

y2 = (Alnx)x^4 = (Ax^4)lnx

Therefore, the second solution y2 is

A(x^4)lnx

You might be interested in
Please Help Me!!!!
ra1l [238]
2. WXZ=YXZ
3.  Reflexive property of equality
(if they are angles)-
4. WXZ=YXZ ~~Reason- Congruent angles have equal number of degrees
(if they are triangles-more likely judging by the problem)-
4. Triangle WXZ=YXZ ~ Reason- SAS Congruence Postulate
5 0
3 years ago
What is 42÷(15-2 by the power of 3<br><img src="https://tex.z-dn.net/?f=42%20%5Cdiv%2815%20-%202%20%7B%7D%5E%7B3%7D%20%29" id="T
tigry1 [53]
Answer: 6

Explanation: first you have to get rid of the 2 power of 3, 2 x 2 x 2 is 8. Then 15-8=7. Then 42➗7 is 6
4 0
3 years ago
Please answer this thank you
exis [7]

Step-by-step explanation:

option 1

option 3

7 0
3 years ago
Which one is the answer?
Ulleksa [173]

Answer:

B

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Please help I’ll mark brainliest! Thank you and have a great day/night!
Lera25 [3.4K]

Answer:

-10

Step-by-step explanation:

You always want to solve the inner function first.

g(-2)=(-2)³+3

=-8+3

=-5

g(-2)=-5 so you can replace g(-2) with -5

Now you just have f(-5)

Solve.

f(-5)=-5-5

f(-5)=-10

7 0
2 years ago
Other questions:
  • Emily made 75% of the basket that she attempted if she made 9 baskets how many attempts did she make
    15·2 answers
  • 4(1-x)+2x=3(x+1)<br><br> solve for x.
    5·1 answer
  • Which is a cheaper per song? <br> a) $12 for 15 songs <br> b) $9 for 10 songs
    15·1 answer
  • Help please I need this immediately thanks.
    11·2 answers
  • Um help pls.........
    8·2 answers
  • Bryanna is thinking of two numbers. Their sum is −7. One of the numbers is 11. Write an
    8·1 answer
  • ASAP please
    15·1 answer
  • Given the equation y=2x+4, what is the value of y when x=5? *
    15·1 answer
  • salma is 1.45 meters tall. at 3p.m she measures the length of a tree's shadow to 29.35 meters. she stands 24.3 meters away from
    11·2 answers
  • Easy question for lots of points
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!