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4 years ago

Prove that if a is a subset of b then int(a) is a subset of int(b) g

Mathematics

1 answer:

ZanzabumX [31]4 years ago

4 0

S in A does not necessary mean that it is in the interior of A since A is an open set.

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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

Bowie used 1/6 cup of rasperries and 1/5 cup of blueberries to make a berry smoothie. How much fruit did he use in total?

guapka [62]

Answer: 11/30

Step-by-step explanation:

5 0

3 years ago

Read 2 more answers

Landon buys some of his clothes second hand. If 40% of his shirts are second hand and he owns 65 shirts, how many of Landon's sh

Leni [432]

Answer:

Step-by-step explanation:

6.5 is 10% of 65, so to find 40% of 65 we multiply 6.5*4=26

4 0

3 years ago

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Which of these systems of linear equations has no solution?

Alika [10]

Answer:

A is correct

Step-by-step explanation:

A has no solution because since the slopes are the same, setting both equations equal to each other will cancel out the slopes and result in 8=16, meaning no solution will make both sides equal to each other.

B does have a solution because of different slopes

C does have a solution because of different slopes

D does have a solution because of different slopes

7 0

3 years ago

Please help please i have to get this done

oksian1 [2.3K]

Answer:

im not sure sure but by the looks of it i think thats 3,6

Step-by-step explanation:

7 0

3 years ago

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