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SCORPION-xisa [38]
3 years ago
9

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

Mathematics
1 answer:
Kitty [74]3 years ago
4 0
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)
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Factor -7x3 + 21x2 + 3x - 9 by grouping. What is the resulting expression?
irga5000 [103]

Answer:

(x-3)(-7x^2+3) is the resulting expression after grouping

Step-by-step explanation:

Given:

-7x^3+21x^2+3x-9

Now taking common -7x^2 from 1st 2 numbers and and 3 common from last we get

-7x^2(x-3)+3(x-3)

Now taking x-3 common from both numbers we get

(x-3)(-7x^2+3) which is the final equation after grouping

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Step-by-step explanation:

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A shape is rotated counterclockwise 110 degrees about the origin. What is another way to describe the rotation?
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Determine the quotient, q(x), and remainder, r(x) when
Vinvika [58]
ANSWER

Quotient:

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Remainder:

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EXPLANATION

The given functions are

f(x) = 12 {x}^{4} + 21 {x}^{3} + 31 {x}^{2} + 21x + 9

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We perform the long division as shown in the attachment.

The quotient is

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