hw3 - Suggested homework questions: Section 3.4, Repeated...

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Unformatted text preview: Suggested homework questions: Section 3.4, Repeated roots, reduction of order (continued) • # 21. Suppose r 1 and r 2 are roots of ar 2 + br + c = 0 and that r 1 6 = r 2 ; then e r 1 t and e r 2 t are solutions of the differential equation ay 00 + by + cy = 0. Show that φ ( t ; r 1 ,r 2 ) = e r 2 t- e r 1 t r 2- r 1 is also a solution of the equation for r 2 6 = r 1 . Then think of r 1 as fixed but use L’Hospitals rule to evalutate the limit φ ( t ; r 1 ,r 2 ) as r 2 → r 1 , thereby obtaining the second solution in the equal roots case. Use the method of reduction to find a second solution of the differential equation: • # 23. t 2 y 00- 4 ty + 6 y = 0 , t > , y 1 ( t ) = t 2 . • # 25. y 1 ( t ) = t- 1 t 2 y 00 + 3 ty + y = 0 t > . • # 27. y 1 ( x ) = sin( x 2 ), xy 00- y + 4 x 3 y = 0 . (Euler equations) In each of the following use the substitution introduced in problem 34 Section 3.3 to solve the equations: • # 41. t 2 y 00- 3 ty + 4 y = 0 t > . • # 45. 4 t 2 y 00- 8 ty + 9 y = 0 , t > . • # 46. t 2 y 00 + 5 ty + 13 y = 0 , t > . Section 3.5, Method of undetermined coefficients In each of the following find the general solution of the differential equation. Hint: Recall if you want to solve a linear differential equation of the form y 00 + p ( t ) y + q ( t ) y = f ( t )+ g ( t ) then one can solve y 00 + p ( t ) y + q ( t ) y = f ( t ) (say the solution is y f ) and solve y 00 + p ( t ) y + q ( t ) y = g ( t ) (say the solution is y g ) and then y = y f + y g solves the original problem. This will make the computation easier. • # 1. y 00- 2 y- 3 y = 3 e 2 t . 1 • # 3. y 00- 2 y- 3 y =- 3 te- t . • # 7. 2 y 00 + 3 y + y = t 2 + 3 sin( t ) . • # 8. y 00 + y = 3 sin(2 t ) + t cos(2 t ) . In each of the following solve the initial value problem: • # 13. y 00 + y- 2 y = 2 t, y (0) = 0 ,y (0) = 1 . • # 15. y 00- 2 y + y = te t + 4 , y (0) = 1 ,y (0) = 1 . • # 16. y 00- 2 y- 3 y = 3 te 2 t , y (0) = 1 ,y (0) = 0 . Determine a suitable form for Y ( t ) if the method of undetermined coefficients is to be used. Hint: Recall if your choice Y ( t ) solves the homogeneous version of the equation then you need to multiply by t and if that still satisfies the homogenous equation then multiply by t 2 . The answers for this one are in back of book. The hint for the above questions also apply here. • # 19. y 00 + 3 y = 2 t 4 + t 2 e- 3 t + sin(2 t ) . • # 21. y 00- 5 y + 6 y = e t cos(2 t ) + e 2 t (3 t + 4) sin( t ) . • # 24. y 00 + 4 y = t 2 sin(2 t ) + (6 t + 7) cos(2 t ) . • # 25. y 00 + 3 y + 2 y = e t ( t 2 + 1) sin(2 t ) + 3 e- t cos( t ) + 4 e t . Section 3.6 Variation of parameters Use the method of variation of parameters to find a particular solution of: • # 1....
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This note was uploaded on 12/29/2010 for the course MATH 263 taught by Professor Coombs during the Spring '08 term at UBC.

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hw3 - Suggested homework questions: Section 3.4, Repeated...

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