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Simora [160]
3 years ago
7

The indicated function y1(x) is a solution of the given differential equation. use reduction of order or formula (5) in section

4.2, y2 = y1(x) e−∫p(x) dx y 2 1 (x) dx (5) as instructed, to find a second solution y2(x). 9y'' − 12y' + 4y = 0; y1 = e2x/3
Mathematics
1 answer:
Ber [7]3 years ago
7 0
Given that y_1=e^{2x/3}, we can use reduction of order to find a solution y_2=v(x)y_1=ve^{2x/3}.

\implies {y_2}'=\dfrac23ve^{2x/3}+v'e^{2x/3}=\left(\dfrac23v+v'\right)e^{2x/3}
\implies{y_2}''=\dfrac23\left(\dfrac23v+v'\right)e^{2x/3}+v''e^{2x/3}=\left(\dfrac49v+v'+v''\right)e^{2x/3}

\implies9y''-12y'+4y=0
\implies 9\left(\dfrac49v+v'+v''\right)e^{2x/3}-12\left(\dfrac23v+v'\right)e^{2x/3}+4ve^{2x/3}=0
\implies9v''-3v'=0

Let u=v', so that

9u'-3u=0\implies 3u'-u=0\implies u'-\dfrac13u=0
e^{-x/3}u'-\dfrac13e^{-x/3}u=0
\left(e^{-x/3}u\right)'=0
e^{-x/3}u=C_1
u=C_1e^{x/3}

\implies v'=C_1e^{x/3}
\implies v=3C_1e^{x/3}+C_2

\implies y_2=\left(3C_1e^{x/3}+C_2\right)e^{2x/3}
\implies y_2=3C_1e^x+C_2e^{2x/3}

Since y_1 already accounts for the e^{2x/3} term, we end up with

y_2=e^x

as the remaining fundamental solution to the ODE.
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<u>Given</u><u> </u><u>Information</u><u> </u><u>:</u><u>-</u>

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  • A polygon with 10 sides ( Decagon )

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<u>To</u><u> </u><u>Find</u><u> </u><u>:</u><u>-</u>

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  • The value of one of the exterior angles

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<u>Formula</u><u> </u><u>Used</u><u> </u><u>:</u><u>-</u>

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\qquad \diamond \:  \underline{ \boxed{ \pink{ \sf Exterior ~angle = \dfrac {360^\circ}{no. ~of~sides}}}} \:  \star

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<u>Solution</u><u> </u><u>:</u><u>-</u>

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Putting the given values, we get,

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\sf \dashrightarrow Exterior ~angle =  \dfrac{360  ^\circ}{10} \:  \:   \\  \\  \\ \sf \dashrightarrow Exterior ~angle =  \frac{36 \cancel{0}^\circ}{ \cancel{10}} \:  \:   \\  \\  \\ \sf \dashrightarrow Exterior ~angle =  \underline{ \boxed{ \frak{ \red{36^\circ}}}} \: \star \\  \\

Thus, the value of the exterior angles of a Decagon is 36°.

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\underline{ \rule{227pt}{2pt}} \\  \\

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Maslowich

y varies directly with x.

The constant of variation is k = 7

Step-by-step explanation:

We have to check whether both vary directly with each other or not.

Every pair of x and y will be considered one by one

If y varies directly with x, it should satisfy y = kx or k = yx

So,

For x= -2, y=-14

-14 = -2k\\k = \frac{-14}{-2}\\k = 7

For

x=3, y=21

21 = 3k\\ k = \frac{21}{3}\\k = 7

For

x=5, y=35

35 = 5k\\ k = \frac{35}{5}\\k = 7

As k is same for all pairs of x and y, it can be concluded that:

y varies directly with x.

The constant of variation is k = 7

Keywords: Proportion, variation

Learn more about proportion at:

  • brainly.com/question/10703930
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#LearnwithBrainly

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