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
o-na [289]
3 years ago
10

Consider the differential equation y'' − y' − 12y = 0. Verify that the functions e−3x and e4x form a fundamental set of solution

s of the differential equation on the interval (−[infinity], [infinity]). The functions satisfy the differential equation and are linearly independent since the Wronskian W e−3x, e4x = $$7ex ≠ 0 for −[infinity] < x < [infinity].
Mathematics
1 answer:
Inga [223]3 years ago
3 0

Answer:

e^(-3x) and e^(4x) form a fundamental set of solutions of the differential equation

y'' - y' - 12y = 0.

Since their wronskian is 7e^x and not 0

Step-by-step explanation:

Given the differential equation

y'' - y' - 12y = 0.

We are required to verify that the functions e^(-3x) and e^(4x) form a fundamental set of solutions of the differential equation on the interval (−[infinity], [infinity]). They form a fundamental set if they are linearly independent, and they are linearly independent if their wronkian is not zero, otherwise, they are linearly dependent.

Now, we need to find the Wronskian of e^(-3x) and e^(4x), and see if it is equal to zero or not.

The wrinkles of two functions y1 and y2 is given as the determinant

W(y1, y2) = |y1...........y2|

......................|y1'.........y2'|

W(e^(-3x), e^(4x))

= |e^(-3x)............e^(4x)|

...|-3e^(-3x)......4e^(4x)|

= e^(-3x) × 4e^(4x) - (-3e^(-3x) × e^(4x))

= 4e^x + 3e^x

= 7e^x

≠ 0

Since the wronskian is not zero, we conclude that the are linearly independent, and hence, form a set of fundamental solution.

You might be interested in
A rectangle is four times as long as it is wide. If its length were diminished by 6 meters and its width were increased by 6 met
valkas [14]

Answer:

Length equals 16 and Width equals 4

Step-by-step explanation:

First let us create an equation. We can use L and W for length and width.

If the length is 4 times the width, then we end up with: L = 4W

It then says, " If its length were diminished by 6 meters and its width were increased by 6 meters, it would be a square."

Since a square has an equal length and width then we end up with:

L - 6 = W + 6

Knowing this we can just substitute the first equation into the second one leaving us with: 4W - 6 = W + 6

We then remove a W from both sides so that the right side is left with a 6, and add 6 to both sides to remove the -6 from the left one.

This leaves us with 3W = 12

W = 4, and if we put that into our first equation, L = 4W, then Length equals 16, and Width equals 4. We can check this by putting it into the 2nd equation. 16 - 6 = 4 + 6.

7 0
2 years ago
Find the x and y intercept of y=7×+5
Zina [86]
X-int. is (5/7,0)
y-int. is (0,5)
3 0
3 years ago
What is g(x)?<br> A g(x)=-x2<br> B g(x)=-2x<br> C g(x)= -x<br> D g(x)= -|x|
boyakko [2]

Answer:

Should be A

Hope this helped!

8 0
3 years ago
Read 2 more answers
Alan, david and shari are running for president of their high school. There were a total of 1,000 votes cast for president. Davi
vekshin1

Answer:

Alan received 350 votes

Step-by-step explanation:

David received 250 votes

Shari received  400 votes

1,000 - 250=750

750 - 400 = 350

7 0
3 years ago
F(x)=4(x-1)^2-1 axis of symmetry
Levart [38]
The vertex of this parabola has coordinates (1,-1).
The axis symmetry goes trough the vertex. The equation of the axis of symmetry is x=1.
6 0
3 years ago
Other questions:
  • 3x+2(4x-4)=3<br> answer choices-1,2,3,4
    12·2 answers
  • The sum of two numbers is 50. One number is 10 less than three times the other number. What are the two numbers? 25 and 25 45 an
    15·1 answer
  • What is the value of x?<br><br> sin(x+22)°=cos(2x−7)°
    10·1 answer
  • Find the sum.<br><br> 17x3 + (3x + 8x3) = <br> (from clever)
    6·2 answers
  • What is 900,000 in scientific notation​
    12·1 answer
  • Mako’s goal is to save $42. He currently has $2 and plans to save $18 per week. Let x represent the number of weeks Mako saves m
    12·1 answer
  • Help me please?/+854​
    12·1 answer
  • PLZZZ HELP 30 POINTS!!!!!!!
    9·1 answer
  • Find the length of AC.<br>a. 10.1 <br>b. 10.0<br>c. 10.9<br>d. 8.86<br> PLEASE HELPPP ASAP ​
    11·1 answer
  • PLZ SOMEONE HELP ME I NEED HELP PLEASEE
    7·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!