A line and a point _______ intersect 1. Always 2. Sometimes 3. Never

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

Study the table representing the price of different amounts of apples, in pounds, and then complete the sentences. The constant

stellarik [79]

Answer: the first part is 1.25. The second part is y=1.25x

Step-by-step explanation: edge 2021

4 0

3 years ago

Find the values of m and b that make the following function differentiable.

dimaraw [331]

Now, bearing in mind that, for a graph, to be "differentiable", means a "smooth transition", not an abrupt edge or a "cusp", the first subfunction of the piece-wise is just a quadratic, the second subfunction is a linear, well, so the linear needs to hit the quadratic "smoothly", not make an abrupt edge, for that, the linear must continue the quadratic from 2 onwards, since 2 is the range edge.

now, what is the piece-wise when x = 2? well, f(2) = x² or (2)² or 4

that means, the linear needs to hit it at that point, to make it so, let's make it the slope of the linear, the same as the quadratic's.

what's the quadratic's slope? well, simple enough

$\bf \left. \cfrac{dy}{dx}=2x \right|_{x=2}\implies 4\impliedby \textit{this means }
\begin{array}{llll}
mx+b\\
\uparrow \\
4
\end{array}$

so we have y = 4x + b

now, recall, for the quadratic, when x =2, y = 4, so for the linear, when x = 2, y = 4 as well, this means

$\bf 4 = 4(2) + b\implies 4=8+b\implies 4-8=b\implies \boxed{-4=b}
\\\\\\
mx+b\implies 4x-4\\\\
-------------------------------\\\\
f(x)=
\begin{cases}
x^2&x\le 2\\
4x-4&x\ \textgreater \ 2
\end{cases}$

check the picture below.

4 0

3 years ago

Find the area of the figure

MakcuM [25]

9514 1404 393

Answer:

24.885 in²

Step-by-step explanation:

Use the formula for the area of a triangle.

A = 1/2bh . . . . . . . where b is the base length and h is the height perpendicular to the base

A = 1/2(7.9 in)(6.3 in) = 24.885 in²

_____

Additional comment

The given side lengths cannot form a triangle, as the side shown as 14.7 is too long for the ends of the other segments to connect to. The attachment shows that side should be 11.97 in.

5 0

3 years ago

Write and solve the equation and then check your answer. Which statements are true about the equation and its solution? Check

ch4aika [34]

The correct equation is n - 31 = 52

To check the solution, substitute 83 for the variable in the equation

This is an addition problem.

To solve the equation, add 31 to both sides.

Step-by-step explanation:

A number decreased by thirty-one is fifty-two

That means:

Subtract 31 from a number
The difference equal 52

Assume that the number is n

∵ The number is n

∵ When n decreased by 31 the answer is 52

∴ The equation is n - 31 = 52

The correct equation is n - 31 = 52

∵ n - 31 = 52

- To find n add 31 to both sides of the equation

∴ n - 31 + 31 = 52 + 31 ⇒ addition property

∴ n = 83

-To check the equation substitute n by 83 in the left hand side, the

answer must be equal the right hand side

∵ Left hand side = 83 - 31 = 52

∵ Right hand side = 52

∴ n = 83 is correct answer

To check the solution, substitute 83 for the variable in the equation

This is an addition problem.

To solve the equation, add 31 to both sides.

Learn more:

You can learn more about solving equation in one variable in brainly.com/question/11306893

#LearnwithBrainly

7 0

3 years ago

Read 3 more answers