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zhenek [66]
3 years ago
5

Consider the differential equation

Physics
1 answer:
Zielflug [23.3K]3 years ago
3 0
Yp(t) = A1 t^2 + A0 t + B0 t e(4t)

=> y ' = 2A1t + A0 + B0 [e^(4t) +4 te^(4t) ]

     y ' = 2A1t + A0 + B0e^(4t) + 4B0 te^(4t)

=> y '' = 2A1 + 4B0e(4t) + 4B0 [ e^(4t) + 4te^(4t)

     y '' = 2A1 + 4B0e^(4t) + 4B0e^(4t) + 16B0te^(4t)

Now substitute the values of y ' and y '' in the differential equation:

 
<span>y′′+αy′+βy=t+e^(4t)


</span> 2A1 + 4B0e^(4t) + 4B0e^(4t) + 16B0te^(4t) + α{2A1t + A0 + B0e^(4t) + 4B0 te^(4t) } + β{A1 t^2 + A0 t + B0 t e(4t)} = t + e^(4t)

Next, we equate coefficients

1) Constant terms of the left side = constant terms of the right side:

2A1+ 2αA0 = 0 ..... eq (1)

2) Coefficients of e^(4t) on both sides

8B0 + αB0 = 1 => B0 (8 + α) = 1 .... eq (2)

3) Coefficients on t

2αA1 + βA0 = 1 .... eq (3)


4) Coefficients on t^2

βA1 = 0 ....eq (4)

given that A1 ≠ 0 => β =0

5) terms on te^(4t)

16B0 + 4αB0 + βB0 = 0 => B0 (16 + 4α + β) = 0 ... eq (5)

Given that B0 ≠ 0 => 16 + 4α + β = 0

Use the value of β = 0 found previously

16 + 4α = 0 => α = - 16 / 4 = - 4.

Answer: α = - 4 and β = 0
  




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Answer:

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Explanation:

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√(R² + X²) = 16 Ω

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X² = 186.6111

X = √186.6111

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Since X = 314.16L = 13.66 Ω

L = 13.66/314.16

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ii. Since the same current is supplied in both circuits, the impedance Z of the circuit is Z = 240 V/15 A = 16 Ω.

So in phasor form Z = (8.33 + j13.66) Ω

iii. The phase difference θ between the current and voltage is  

θ = tan⁻¹X/R

= tan⁻¹(314.16L/R)

= tan⁻¹(314.16 × 0.0435 H/8.33 Ω)

= tan⁻¹(13.66/8.33)

= tan⁻¹(1.6406)

= 58.64°

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Will a beam of light that is, at first, in air and oriented perpendicular to the surface of a body of water be deflected as a re
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