Query 20

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course MTH 279

4/11 9:48 am

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: 

 y_C = c1 cos t + c2 sin t

y_P= u1*(cos t) + u2*(sin t)

Effective derivatives:

y'= u1'*cos t + u2'* sin t

y''= u1'*(-sin t) + u2'*(cos t)

system:

[[cos t, sin t][-sin t, cos t]]

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

[u1', u2']' = [[cos t, -sin t][-cos t, sin t]]*[0, sec t]'

u1'= -sin t * sec t = -tan t --> u1 = ln(cos t)

u2'= cos t * sec t = 1 --> u2 = t

y= c1*cos t+ c2*sin t + ln(cos t)*cos t + t sin t

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Given Solution: 

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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: 

 y_C= c1cos 6t + c2 sin 6t

y_P= u1cos 6t + u2 sin 6t

y'= u1' cos 6t + u2'sin 6t

y''= u1'(-6 sin 6t) + u2' (6 cos 6t)

[[cos 6t, sin 6t][-6sin 6t, 6 cos 6t]]

W= 6 cos^2 6t + 6 sin^2 6t= 6

u1'= -sin 6t* csc^3 (6t) *1/6= -1/6 csc^2 6t

u1= 1/36*cot 6t

u2'= 1/6 cos 6t * csc^3 6t = 1/6 * cot 6t * csc^2 6t

u2= 1/72 cot^2 6t

y= c1 cos 6t + c2 sin 6t -1/36 csc^2 6t + 1/72 cot^2 6t 

confidence rating #$&*:

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Given Solution: 

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Self-critique (if necessary):

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Self-critique rating:"

Self-critique (if necessary):

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Self-critique rating:

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