2065s10ex1a

2065s10ex1a - Name: Solutions Exam 1 Instructions. Answer...

This preview shows pages 1–2. Sign up to view the full content.

Name: Solutions Exam 1 Instructions. Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without supporting work. Put your name on each page of your paper. A table of Laplace transforms and a short table of integrals are appended to the exam. 1. [17 Points] Find the general solution of: y 0 = 6 t ( y ¡ 1) 2 = 3 . I Solution. This equation is separable. After separating variables, it becomes ( y ¡ 1) ¡ 2 = 3 y 0 = 6 t , which in diﬁerential form is ( y ¡ 1) ¡ 2 = 3 dy = 6 tdt and integration then gives 3( y ¡ 1) 1 = 3 = 3 t 2 + C . Dividing by 3 gives ( y ¡ 1) 1 = 3 = t 2 + K , where K = C= 3 is an arbitrary constant. Cubing both sides and solving for y gives y = ( t 2 + K ) 3 + 1 : J 2. [17 Points] Find the general solution of: y 0 ¡ 4 y = 3 e 4 t + 4 e 3 t . I Solution. This equation is linear with coe–cient function p ( t ) = ¡ 4 so that an integrating factor is given by ( t ) = e R ¡ 4 dt = e ¡ 4 t . Multiplication of the diﬁerential equation by the integrating factor gives e ¡ 4 t y 0 ¡ 4 e ¡ 4 t y = 3 + 4 e ¡ t ; and the left hand side is recognized (by the choice of ( t )) as a perfect derivative: d dt ( e ¡ 4 t y ) = 3 + 4 e ¡ t : Integration then gives e ¡ 4 t y = 3 t ¡ 4 e ¡ t + C; where C is an integration constant. Multiplying through by e 4 t gives y = 3 te 4 t ¡ 4 e 3 t + Ce ¡ 4 t : J 3. [17 Points] Solve the initial value problem: y 0 + 6 t y = 11 t 4 , y (1) = 3.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

2065s10ex1a - Name: Solutions Exam 1 Instructions. Answer...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online