Winter 2006 - Reynold's Class - Exam 2 (Version B)

# Winter 2006 - Reynold's Class - Exam 2 (Version B) - Exam 2...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
1. [ 25 points ] (Variation of Parameters) Given the two linearly independent solutions y 1 ( t ) = t, y 2 ( t ) = te t , to the linear, second-order homogeneous equation, t 2 y - t ( t + 2) y + ( t + 2) y = 0 , t > 0 , determine the general solution to the non-homogeneous ordinary differential equation, t 2 y - t ( t + 2) y + ( t + 2) y = t 3 e 3 t , t > 0 . Solution: In order to use the variation of parameters method, the differential equation must be in the form y + p ( t ) y + q ( t ) y = g ( t ) . Since t > 0 , we may divide every term by t 2 to convert the equation to the proper form, y - t + 2 t y + t + 2 t 2 y = te 3 t , Thus our non-homogeneous term is g ( t ) = te 2 t (5 pts for using the correct g ( t ) ). Given the two solutions y 1 ( t ) = t , y 2 ( t ) = te t , the Wronskian is given by (3 pts for correct Wronskian) W ( y 1 , y 2 )( t ) = y 1 y 2 - y 1 y 2 = te t + t 2 e t - te t = t 2 e t . We now have all of the necessary components to use the variation of parameters formula for the particular solution Y ( t ) (7 pts for correct setup, 5 points for correct Y ( t ) ): Y ( t ) = - y 1 ( t ) y 2 ( t ) g ( t ) W ( y 1 , y 2 )( t ) dt + y 2 ( t ) y 1 ( t ) g ( t ) W ( y 1 , y 2 )( t ) dt (1) = - t te t ( te 3 t ) t 2 e t dt + te t t ( te 3 t ) t 2 e t dt (2) = - t e 3 t dt + te t e 2 t dt (3) = - t 1 3 e 3 t + te t 1 2 e 2 t (4) = - t 3 e 3 t + t 2 e 3 t (5) = t 6 e 3 t . (6) The general
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern