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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.

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What are the explicit equation and domain for an arithmetic
max2010maxim [7]

Given:

First term of an arithmetic sequence = 5

Second term = 3

To find:

The explicit formula for the given arithmetic sequence.

Solution:

We have,

First term: a_1=5

Second term: a_2=3

Common difference is

d=a_2-a_1

d=3-5

d=-2

Now, the explicit formula for an arithmetic sequence is

a_n=a+(n-1)d

where, a is first term and d is common difference.

Putting a=5 and d=-2, we get

a_n=5+(n-1)(-2)

It can also be written as

a_n=5-2n+2

a_n=7-2n

Here, n is an integer greater than or equal to 1.

Domain is the set of input values.

Therefore, the explicit equation is a_n=5+(n-1)(-2) or a_n=7-2n and domain is all interest greater than or equal to 1.

8 0
2 years ago
Which expressions are equivalent to -2(5x-1)+7(5x-1)-3(-5x+1)
blondinia [14]

Answer:

x=1/5

Step-by-step explanation:

To solve this question we will have to open the bracket first

So let's go back to the given question

-2(5x-1)+7(5x-1)-3(-5x+1)

We can actually equate the equation to zero

Let's solve

-10x+2+35x-7+15x-3=0

Let's collect like terms

-10x+35x+15x=-2+7+3

40x=8

Let's make x the subject of formula by dividing both sides by 40

x=1/5

So the final answer is 1/5

8 0
2 years ago
(Find the distance between the points A(4, 3) and B(7,-1).​
lana66690 [7]

Answer:

5

Step-by-step explanation:

3 0
3 years ago
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What’s 16.0+8.49 <br> By showing work?
natka813 [3]

Answer:

24.49

Step-by-step explanation:

16.00

8.49

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24.49

6 0
2 years ago
Any one that knows this please help thanks : )
Olin [163]

Replace X with -2 then solve:

(-2) +3 / (-2)-1

1 / -3 which is rewritten as -1/3

4 0
3 years ago
Read 2 more answers
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