Query 20

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course MTH 279

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:

λ^2 + 1 = 0

λ = ±i

y_1 = e^(-it)

y_2 = e^(it)

y_c = Ae^(-it) + Be^(it) = (A+B)cos(t) + (B-A)sin(t)

y_p = u_1*cos(t) + u_2*sin(t)

We impose that u_1’*cos(t) + u_2’*sin(t) = 0 in the next step.

y_p’ = [-u_1*sin(t) + u_2*cos(t)] + [u_1’*cos(t) + u_2’*sin(t)] = -u_1*sin(t) + u_2*cos(t)

y_p” = -u_1*cos(t) - u_2*sin(t) - u_1’*sin(t) + u_2’*cos(t)

Plugging in and simplifying:

u_1’*sin(t) + u_2’*cos(t) = sec(t)

[y_1,y_2; y_1’,y_2’]*[u_1’; u_2’] = [0;sec(t)]

[u_1’;u_2’] = [-y_2*sec(t); y_1*sec(t)]

ʃ[u_1’;u_2’]dt = ʃ[-y_2*sec(t); y_1*sec(t)]dt

[u_1; u_2] = [ln(cos(t))+C; t+D]

y_p = (ln(cos(t))+C)*cos(t) + (t+D)*sin(t)

y_g = (A+B)cos(t) + (B-A)sin(t) + ln(cos(t))*cos(t) + Ccos(t) + t*sin(t) + Dsin(t)

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK

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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:

λ^2 + 36 = 0

λ = ±6i

y_1 = e^(-6it)

y_2 = e^(6it)

y_c = e^(-6it) + e^(6it) = (A+B)cos(6t) + (B-A)sin(6t)

y_p = u_1*cos(6t) + u_2*sin(6t)

We impose that u_1’cos(6t) + u_2’sin(6t) = 0 in the next step.

y_p’ = [-6u_1sin(6t) + 6u_2cos(6t)] + [u_1’cos(6t) + u_2’sin(6t)] = -6u_1sin(6t) + 6u_2cos(6t)

y_p” = -36u_1cos(6t) - 36u_2sin(6t) - 6u_1’sin(6t) + 6u_2’cos(6t)

Plugging into the DE gives

-6u_1’sin(6t) + 6u_2’cos(6t) = csc^3(6t)

[cos(6t),sin(6t); -6sin(6t), 6cos(6t)]*[u_1’;u_2’] = [0;csc^3(6t)]

W(t) = 6

[u_1’;u_2’] = [cos(6t), -sin(6t)/6; sin(6t), cos(6t)/6]*[0;csc^3(6t)]

[u_1’; u_2’] = [-sin(6t)/6*csc^3(6t); cos(6t)/6*csc^3(6t)]

ʃ[u_1’; u_2’]dt = ʃ[-sin(6t)/6*csc^3(6t); cos(6t)/6*csc^3(6t)]dt

u_1 = 1/36*cot(6t)+C

u_2 = -1/72*csc^2(6t) + D

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

y_g = (A+B)cos(6t) + (B-A)sin(6t) + 1/36*cot(6t)+C)*cos(6t) + (-1/72*csc^2(6t) + D)*sin(6t)

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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Given Solution:

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Self-critique (if necessary): OK

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Self-critique rating: OK"

Self-critique (if necessary):

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Self-critique rating:

Self-critique (if necessary):

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Self-critique rating:

#*&!

&#Very good responses. Let me know if you have questions. &#