Query_20

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course Mth 279

3 / 31 14

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:

A complimentary equation of

y^2 + 1 = 0

y = +/ - i

y_c = c_1 cos(t) + c_2 sin(t)

y_1 = cos (t) , y_2 = sin (t)

y_p = y_1 u_1 + y_2 u_2

y_pТ = y_1 u_1Т + y_2 u_2Т

y_pТТ = y_1Т u_1Т + y_2Т u_2Т

W = | cos(t) sin(t) , - sin(t) cos(t) |

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

W(t) * [u_1Т , u_2Т] = [ 0 , g ] = [ 0 , sec(t)]

Inver the Wronksian for the following equation

u_1Т = - sin(t) sec(t) = - sin(t) ( 1 / cos(t) ) = tan(t)

u_1 = int tan(t) = ln sec(t)

u_2Т = cos(t) sec(t) = cos(t) ( 1 / cos(t)) = 1

u_2 = t

y_p = cos(t) ln sec(t) + t sin(t)

y = c_1 cos(t) + c_2 sin(t) + cos(t) ln sec(t) + t sin(t)

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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:

y^2 + 36 = 0

y = +/- 6i

y_c = c_1 cos(6 t) + c_2 sin(6 t)

y_p = y_1 u_1 + y_2 u_2

y_pТ = u_1Т cos(6 t) + u_2Т sin(6 t)

y_pТТ = u_1Т (-6 sin(6 t)) + u_2Т (6 cos(6 t)

W = | cos(6 t) sin(6 t) , - 6 sin(6 t) 6 cos(6 t)|

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

u_1Т = - sin(6 t) * csc^3(6 t) * (1 / 6) = - 1 / 6 csc^2 (6 t)

u_1 = 1 / 36 cot(6 t)

u_2Т = cos(6 t) * csc^3(6 t) * ( 1 / 6) = 1 / 6 cot (6 t) csc^2(6 t)

u_2 = 1 / 72 cot^2(6 t)

y = c_1 cos(6 t) + c_2 sin(6 t) - 1 / 36 csc^2(6 t) + 1 / 72 cot^2(6 t)