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STALIN [3.7K]
2 years ago
13

Find the value of y for a given value of x, if y varies directly with x.

Mathematics
1 answer:
lesantik [10]2 years ago
6 0

Answer:

The value of y is 92 . The complete steps of the solution is provided below.

Step-by-step explanation:

We are given with

* y varies directly with X

* If y=128 when x=64.

The equation for direct variation equation is  y= k x.

Where 'k' is constant of variation.

By putting the values of x and y, we can find the value for k.

→ 128 = k (64)

Divide both sides by 64.

→ \frac{128}{64} = k

Simplify it.

→ 2= k  or k =2

Plug in back 'k' as 2 into the equation y=k x.

We get y=2 x.

We are asked to find the value y when x=46.

Plug in x as 46 into the equation.

⇒ y=2(46)

⇒ y= 92

Thus the answer is 92.

Learn more:

brainly.com/question/472621

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\underbrace{y(7xy^2+6)}_{M(x,y)}\,\mathrm dx+\underbrace{x(xy^2-1)}_{N(x,y)}\,\mathrm dy=0

For the ODE to be exact, we require that M_y=N_x, which we'll verify is not the case here.

M_y=21xy^2+6
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So we distribute an integrating factor i(x,y) across both sides of the ODE to get

iM\,\mathrm dx+iN\,\mathrm dy=0

Now for the ODE to be exact, we require (iM)_y=(iN)_x, which in turn means

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Renaming M(x,y) and N(x,y) to our current coefficients, we end up with partial derivatives

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as desired, so our new ODE is indeed exact.

Next, we're looking for a solution of the form \Psi(x,y)=C. By the chain rule, we have

\Psi_x=7x^6y+6x^5y^{-1}\implies\Psi=x^7y+x^6y^{-1}+f(y)

Differentiating with respect to y yields

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\implies\dfrac{\mathrm df}{\mathrm dy}=0\implies f(y)=C

Thus the solution to the ODE is

\Psi(x,y)=x^7y+x^6y^{-1}=C
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