Query_20

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course Mth 279

5/10

Query 20 Differential Equations*********************************************

Question: Using variation of parameters, solve the equation

y '' + y = sec(t), -pi/2 < t < pi/2.

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Your solution:

P(a) = a^2 + 1

a_1,2 = +- i

y_c(t) = Asin(t) + Bcos(t)

assuming that y_p(t) = sin(t)u1(t) + cos(t)u2(t)

and sin(t)u1'(t) + cos(t)u2'(t) = 0

[sin(t) cos(t)] [u1'(t)] = [ 0 ]

[cos(t) -sin(t)] [u2'(t)] [sec(t) ]

W(t) = -sin^2(t) - cos^2(t) = -1

u1'(t) = -y2(t)g(t)/W(t)

= -cos(t)sec(t)/-1 = cos(t)sec(t) = cos(t)(1/cos(t)) = 1

u2'(t) = y1(t)g(t)/W(t) = sin(t)sec(t)/-1 = -sin(t)sec(t) = -sin(t)(1/cos(t)) = -tan(t)

[u1'(t)] = [ 1 ]

[u2'(t)] [-tan(t)]

u1(t) = Int(1)dt = t + c

u2(t) = -Int(tan(t))dt = ln(cos(t)) + c2

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y_c(t) = Asin(t) + Bcos(t)

y_p(t) = tsin(t) + ln(cos(t))cos(t)

y(t) = Asin(t) + Bcos(t) + tsin(t) + ln(cos(t))cos(t)

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):

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Self-critique rating:

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Question: Using variation of parameters, solve the equation

y '' + 36 y = csc^3 ( 6 t ).

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Your solution:

First solving for the homogeneous solution:

P(a) = a^2 + 36 = 0

y_c(t) = Asin(6t) + Bcos(6t)

Then assuming y_p(t) = sin(6t)u1(t) + cos(6t)u2(t)

and sin(6t)u1'(t) + cos(6t)u2'(t) = 0

[sin(6t) cos(6t)] [u1'(t)] = [ 0 ]

[6cos(6t) -6sin(6t)] [u2'(t)] [csc^3(6t) ]

W(t) = -6sin^2(6t) - 6cos^2(6t)

= -6(sin^2(6t) + cos^2(6t))

= -6

u1'(t) = -y2(t)g(t)/W(t)

= -cos(6t)csc^3(6t)/-6 = -cos(6t)/(-6sin^3(6t)) = cot(6t)csc^2(6t)/6

u2'(t) = y1(t)g(t)/W(t)

= sin(6t)csc^3(6t)/-6 = sin(6t)/(-6sin^3(6t)) = -csc^2(6t)/6

[u1'(t)] = [cot(6t)csc^2(6t)/6]

[u2'(t)] [-csc^2(6t)/6]

u1(t) = (1/6)Int(cot(6t)csc^2(6t))dt = (-1/72)cot^2(6t) + c

u2(t) = (-1/6)Int(csc^2(6t))dt = (1/36)cot(6t) + c2

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y_c(t) = Asin(6t) + Bcos(6t)

y_p(t) = (-1/72)cot^2(6t) sin(t) + (1/36)cot(6t) cos(t)

y_p(t) = (-1/72)(cos^2(6t)/sin^2(6t)) sin(t) + (1/36)(cos(6t)/sin(6t)) cos(t)

y_p(t) = (-1/72)(cos^2(6t)/sin(6t)) + (1/36)(cos^2(6t)/sin(6t))

y_p(t) = (-1/72)(cos(6t)cot(6t)) + (1/36)(cos(6t)cot(6t))

y_p(t) = (1/72)(cos(6t)cot(6t))

y(t) = Asin(6t) + Bcos(6t) + (1/72)cos(6t)cot(6t)