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3 years ago

Is every linear relationship a direct variation. Explain please

1 answer:

3 0

Direct variation: y=kx where k is some number thats not 0

that means if x is 2, y is 2k. if x is 1, y is k. if x is 0, y is 0.

notice: no matter what k is, when you put 0 in for x, y will always be 0. that means that a direct variation always goes through the origin, a.k.a. (0,0).

do all lines go through the origin? nah. a line is y=mx+b, where m and b are some numbers. a line is only a direct variation if b is 0. this makes sense because if b is 0, then the y-intercept is 0, so the line goes through the origin.

in conclusion, no, not all linear relationships are direct varations, but all direct variations are linear relationships.

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Consider the differential equation x^2 y''-xy'-3y=0. If y1=x3 is one solution use redution of order formula to find a second lin

Anastasy [175]

Suppose $y_2(x)=y_1(x)v(x)$ is another solution. Then

$\begin{cases}y_2=vx^3\\{y_2}'=v'x^3+3vx^2//{y_2}''=v''x^3+6v'x^2+6vx\end{cases}$

Substituting these derivatives into the ODE gives

$x^2(v''x^3+6v'x^2+6vx)-x(v'x^3+3vx^2)-3vx^3=0$

$x^5v''+5x^4v'=0$

Let $u(x)=v'(x)$ , so that

$\begin{cases}u=v'\\u'=v''\end{cases}$

Then the ODE becomes

$x^5u'+5x^4u=0$

and we can condense the left hand side as a derivative of a product,

$\dfrac{\mathrm d}{\mathrm dx}[x^5u]=0$

Integrate both sides with respect to $x$ :

$\displaystyle\int\frac{\mathrm d}{\mathrm dx}[x^5u]\,\mathrm dx=C$

$x^5u=C\implies u=Cx^{-5}$

Solve for $v$ :

$v'=Cx^{-5}\implies v=-\dfrac{C_1}4x^{-4}+C_2$

Solve for $y_2$ :

$\dfrac{y_2}{x^3}=-\dfrac{C_1}4x^{-4}+C_2\implies y_2=C_2x^3-\dfrac{C_1}{4x}$

So another linearly independent solution is $y_2=\dfrac1x$ .

3 0

3 years ago

Ava, Kate, and Jin bring all the food to a picnic. Ava brings 34, Kate brings 16, and Jin brings 112. Enter a numerical expressi

juin [17]

Answer:

Step-by-step explanation:

Given that:

Ava bring 34 food to the picnic

Kate bring 16 food to the picnic

JIn brings 112 food to the picnic

To enter these digits by applying equivalent fractions with a denominator of 12;

Their respective numerical expression is:

Ava = $\dfrac{34}{12}$

Ava brought $2 \dfrac{5}{6}$ food to the picnic

Kate = $\dfrac{16}{12}$

Kate brought $1 \dfrac{1}{3}$ food to the picnic

Jin = $\dfrac{112}{12}$

Jin brought $9 \dfrac{1}{3}$ food to the picnic

6 0

3 years ago

Read 2 more answers

assume that mrs.giang's car travels 14 miles on each gallon of gas. if she travels to visit her niece who lives 133 miles away,

zalisa [80]

Answer:

9.5 gallons of gas

Step-by-step explanation:

Given that:

Each gallon gives 14 miles

Mrs giang wants to travel 133 miles

gallons she would need = 133 / 14 miles

gallons she would need = 9.5 gallons

So Mrs.giang will need 9.5 gallons of gas to visit her niece.

i hope it will help you!

8 0

4 years ago

Which describes the correlation shown in the scatterplot?

Alex_Xolod [135]

Answer: C) There is no positive or negative correlation

A positive correlation is one where the values are increasing, and the line of dots are clearly going down to up.

A negative correlation is the opposite. The values are decreasing, and the line goes from up to down.

This correlation is one that is neither positive or negative. It is a straight line where the values neither increase nor decrease.

Hope this helps comment below for more questions :)

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3 years ago

Read 2 more answers

Fill in the blank with a constant, so that the resulting expression can be factored as the product of two linear expressions: 2a

Vitek1552 [10]

Answer:

$2ab - 6a + 5b - 15$