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Mazyrski [523]
3 years ago
15

Consider the differential equation x2y′′ − 9xy′ + 24y = 0; x4, x6, (0, [infinity]). Verify that the given functions form a funda

mental set of solutions of the differential equation on the indicated interval. The functions satisfy the differential equation and are linearly independent since W(x4, x6) = ≠ 0 for 0 < x < [infinity]. Form the general solution. y =
Mathematics
1 answer:
pantera1 [17]3 years ago
5 0

Answer:

The functions satisfy the differential equation and linearly independent since W(x)≠0

Therefore the general solution is

y= c_1x^4+c_2x^6

Step-by-step explanation:

Given equation is

x^2y'' - 9xy+24y=0

This Euler Cauchy type differential equation.

So, we can let

y=x^m

Differentiate with respect to x

y'= mx^{m-1}

Again differentiate with respect to x

y''= m(m-1)x^{m-2}

Putting the value of y, y' and y'' in the differential equation

x^2m(m-1) x^{m-2} - 9 x m x^{m-1}+24x^m=0

\Rightarrow m(m-1)x^m-9mx^m+24x^m=0

\Rightarrow m^2-m-9m+24=0

⇒m²-10m +24=0

⇒m²-6m -4m+24=0

⇒m(m-6)-4(m-6)=0

⇒(m-6)(m-4)=0

⇒m = 6,4

Therefore the auxiliary equation has two distinct and unequal root.

The general solution of this equation is

y_1(x)=x^4

and

y_2(x)=x^6

First we compute the Wronskian

W(x)= \left|\begin{array}{cc}y_1(x)&y_2(x)\\y'_1(x)&y'_2(x)\end{array}\right|

         = \left|\begin{array}{cc}x^4&x^6\\4x^3&6x^5\end{array}\right|

         =x⁴×6x⁵- x⁶×4x³    

        =6x⁹-4x⁹

        =2x⁹

       ≠0

The functions satisfy the differential equation and linearly independent since W(x)≠0

Therefore the general solution is

y= c_1x^4+c_2x^6

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Answer:

x+y/4 = 1/2

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move variables to one side:

multiply the first equation by 4 to get: x+y =2

and the second equation by 3 to get: x-3y =6

then subtract the equations to cancel out x:

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- x-3y = 6

then u get

y--3y = 2-6

4y = -4

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substitute to solve for x:

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x-1 = 2

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3+-1

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correct!!!

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The diameter of circles A,C and E are 32 cm, 24cm and 14 cm respectively
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Answer:

AG = 4

AH = 21

EC = 12

CH = 5

HE = 7

Step-by-step explanation:

<u><em>The complete question is</em></u>

The diameters of circles A, C and E are 32 cm, 24 cm and 14 cm respectively.

Which of the following statements are true? Select all that apply.

•AG = 4

•GC = 10

•AH = 21

•EC = 12

•EH = 5

•CH = 5

•HE = 7

The picture of the question in the attached figure

<u><em>Verify each statement</em></u>

1) AG = 4

we know that

AG=AC-GC

AC=32\2=16\ cm ----> radius of circle A

GC=24/2=12\ cm ----> radius of circle C

substitute

AG=16-12=4\ cm

therefore

The statement is true

2) GC = 10

we know that

GC=24/2=12\ cm ----> radius of circle C

therefore

The statement is false

3) AH = 21

we know that

AH=AC+CH

we have

AC=16\ cm ----> radius of circle A

CH=CE-HE

CE=12\ cm ----> radius of circle C

HE=14/2=7\ cm ----> radius of circle E

so

CH=12-7=5\ cm

AH=16+5=21\ cm

therefore

The statement is true

4) EC = 12

we know that

EC=24/2=12\ cm ----> radius of circle C

therefore

The statement is true

5) EH = 5

we know that

EH=14/2=7\ cm ----> radius of circle E

therefore

The statement is false

6) CH = 5

we know that

CH=CE-HE

CE=12\ cm ----> radius of circle C

HE=14/2=7\ cm ----> radius of circle E

so

CH=12-7=5\ cm

therefore

The statement is true

7) HE = 7

we know that

HE=14/2=7\ cm ----> radius of circle E

therefore

The statement is true

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Jlenok [28]
19.5 meters you multiply 3 seconds by 6.5, there are 6.5 meters per second, thus implying each second is 6.5 meters. Equaling 19.5
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