The circumference of a circle is 25 m. What is the area, in square meters?

Solve the given initial-value problem. x^2y'' + xy' + y = 0, y(1) = 1, y'(1) = 8

Kitty [74]

Substitute $z=\ln x$ , so that

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\mathrm dy}{\mathrm dz}\cdot\dfrac{\mathrm dz}{\mathrm dx}=\dfrac1x\dfrac{\mathrm dy}{\mathrm dz}$

$\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1x\dfrac{\mathrm dy}{\mathrm dz}\right]=-\dfrac1{x^2}\dfrac{\mathrm dy}{\mathrm dz}+\dfrac1x\left(\dfrac1x\dfrac{\mathrm d^2y}{\mathrm dz^2}\right)=\dfrac1{x^2}\left(\dfrac{\mathrm d^2y}{\mathrm dz^2}-\dfrac{\mathrm dy}{\mathrm dz}\right)$

Then the ODE becomes

$x^2\dfrac{\mathrm d^2y}{\mathrm dx^2}+x\dfrac{\mathrm dy}{\mathrm dx}+y=0\implies\left(\dfrac{\mathrm d^2y}{\mathrm dz^2}-\dfrac{\mathrm dy}{\mathrm dz}\right)+\dfrac{\mathrm dy}{\mathrm dz}+y=0$
$\implies\dfrac{\mathrm d^2y}{\mathrm dz^2}+y=0$

which has the characteristic equation $r^2+1=0$ with roots at $r=\pm i$ . This means the characteristic solution for $y(z)$ is

$y_C(z)=C_1\cos z+C_2\sin z$

and in terms of $y(x)$ , this is

$y_C(x)=C_1\cos(\ln x)+C_2\sin(\ln x)$

From the given initial conditions, we find

$y(1)=1\implies 1=C_1\cos0+C_2\sin0\implies C_1=1$
$y'(1)=8\implies 8=-C_1\dfrac{\sin0}1+C_2\dfrac{\cos0}1\implies C_2=8$

so the particular solution to the IVP is

$y(x)=\cos(\ln x)+8\sin(\ln x)$

4 0

2 years ago

Simplify : 14z + 8y - 5 + 9y - 10z

Ugo [173]

Answer:

17y+4z-5

Step-by-step explanation:

6 0

2 years ago

Read 2 more answers

How can one determine the zeros of a function?

OLEGan [10]

<span>They are the values of 'x' that make the function zero.

If you graph the function, they are the values of 'x' where the graph
crosses the x-axis.</span>

8 0

3 years ago

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Need word problem solved A washer and a dryer cost 821 combined. The washer costs 79 less than the dryer. What is the cost of th

Leni [432]

Let x be the cost of dryer, y be the cost of washer

x + y =821
x-79 = y

combine 2 equation
we got
x + (x-79) = 821
2x + 79 = 821
2x = 742
x=471

The cost of dryer is $471

3 0

2 years ago

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The measure of an exterior angle of a regular polygon is given below. Find the measure of an interior angle. Then find the numbe

FromTheMoon [43]

Answer:

Internal angle = $160^\circ$