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Archy [21]
3 years ago
7

Y''+Y'+Y=1 solve ordinary differential equation

Mathematics
1 answer:
taurus [48]3 years ago
5 0

Answer:

y(t) = c₁ e^(-1/2 t) cos(√3/2 t) + c₂ e^(-1/2 t) sin(√3/2 t) + 1

Step-by-step explanation:

y" + y' + y = 1

This is a second order nonhomogenous differential equation with constant coefficients.

First, find the roots of the complementary solution.

y" + y' + y = 0

r² + r + 1 = 0

r = [ -1 ± √(1² − 4(1)(1)) ] / 2(1)

r = [ -1 ± √(1 − 4) ] / 2

r = -1/2 ± i√3/2

These roots are complex, so the complementary solution is:

y = c₁ e^(-1/2 t) cos(√3/2 t) + c₂ e^(-1/2 t) sin(√3/2 t)

Next, assume the particular solution has the form of the right hand side of the differential equation.  In this case, a constant.

y = c

Plug this into the differential equation and use undetermined coefficients to solve:

y" + y' + y = 1

0 + 0 + c = 1

c = 1

So the total solution is:

y(t) = c₁ e^(-1/2 t) cos(√3/2 t) + c₂ e^(-1/2 t) sin(√3/2 t) + 1

To solve for c₁ and c₂, you need to be given initial conditions.

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-5.9, -23/4, 5.78, 58%,
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Answer:

They can W and W.

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Select the correct answer from each drop-down menu. Quadrilateral PQRS, with vertex P(-5, -3), undergoes a transformation to for
anygoal [31]

Answer:

i) A. 180º rotation about the origin, ii) Q' = (4, 5).

Step-by-step explanation:

i) In this case, we understand that vertex P = (-5,-3) changed to P' = (5,3) after doing an operation. At first we must calculate the distance of each point regarding origin by Pythagorean Theorem:

Point P:

OP = \sqrt{(x_{P}-x_{O})^{2}+(y_{P}-y_{O})^{2}}

If we know that x_{P} = -5, y_{P} = -3, x_{O} = 0 and y_{O} = 0, the distance OP is:

OP = \sqrt{(-5-0)^{2}+(-3-0)^{2}}

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Point P':

OP' = \sqrt{(x_{P'}-x_{O})^{2}+(y_{P'}-y_{O})^{2}}

If we know that x_{P'} = 5, y_{P'} = 3, x_{O} = 0 and y_{O} = 0, the distance OP' is:

OP' = \sqrt{(5-0)^{2}+(3-0)^{2}}

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As OP = OP', origin is the center of rotation.

Besides, P' is a multiple of P, that is:

1) (-5, -3) Given

2) ((-1)\cdot 5, (-1)\cdot 3) (-a)\cdot b = -a\cdot b

3) (-1)\cdot (5, 3) Scalar multiplication of a vector/Result.

The value of the scalar proves that P experimented a 180º rotation about the origin. Hence, the correct answer is A.

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Q' = (-1)\cdot Q

Q' = (-1) \cdot (-4,-5)

Q' = ((-1)\cdot (-4), (-1)\cdot (-5))

Q' = (4, 5)

Which corresponds to option C.

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