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dezoksy [38]
3 years ago
15

Use the method of reduction of order to find a second independent solution of the given differential equation. t2y'' + 3ty' + y

= 0, t > 0; y1(t) = t−1
Mathematics
1 answer:
Charra [1.4K]3 years ago
3 0

I assume you mean y_1=t^{-1}, and not y_1=t-1, since this doesn't satisfy the ODE.

Assume a second solution of the form y_2=vy_1, where v is a function of t. Then

{y_2}'=v'y_1+v{y_1}'

{y_2}''=v''y_1+2v'{y_1}'+v{y_1}''

Substituting into the ODE gives

t^2\left(\dfrac{v''}t-\dfrac{2v'}{t^2}+\dfrac{2v}{t^3}\right)+3t\left(\dfrac{v'}t-\dfrac v{t^2}\right)+\dfrac vt=0

\implies tv''+v'=0

\implies(tv')'=0

\implies tv'=C

\implies v'=\dfrac Ct

\implies v=C_1\ln|t|+C_2

\implies y_2=\dfrac{\ln t}t

where we omit the second term because it's already accounted for by y_1.

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Answer:

171

Step-by-step explanation:

Q~r

find k (k is the constant)

Q=kr

76=k×20

76=20k divide 20 on both sides giving you

k=3.8

then Q=k×r

Q=3.8×45

Q=171

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Firdavs [7]

Answer:

Emma spend 8 hours in homework in a week and Jared  spends 16 hours in homework in 2 weeks.

Step-by-step explanation:

Two quantities P and Q are said to be in PROPORTION if and only if for all values of P and q, we get:

\frac{P}{Q}  = k

Here, k : PROPORTIONALITY CONSTANT

Case: 1

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So, the piano practice in 2 weeks( 1 x 2) =   2 x  5 = 10 hours

But here, it is given to be 12 hours in 2 weeks.

Hence, NOT PROPORTIONAL.

Case 2:

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So, the words typed in 210 seconds ( 60 x 3.5) =   70  x  3.5 = 240 words

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Case: 3

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So, the homework done in 2 weeks( 1 x 2) =   2 x 8 = 16 hours

It is given to be 16 hours in 2 weeks.

Hence,  PROPORTIONAL.

Case: 4

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So, the distance covered in 102 minutes (78 x 1.3) =  15 x 1.3 = 19.6 miles

But here, it is given to be 17 miles in 102 minutes.

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Case: 5

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So, the distance covered in 55 minutes (45 x 1.2) =  18 x 1.2  = 21.6 miles

But here, it is given to be 22 miles in 55 minutes.

Hence, NOT PROPORTIONAL.

Hence, the only proportional case is Emma spend 8 hours in homework in a  week and Jared  spends 16 hours in homework in 2 weeks.

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nadezda [96]

Answer:

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Step-by-step explanation:

6(x-4) .....

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