math1005test2sol

# math1005test2sol - 40 marks-50 minutes MATH1005A Summer2009...

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

MATH1005A, Summer2009 Test2 Problem 1 . Use the method of undetermined coeﬃcients or some modiﬁcation of it to ﬁnd general solutions of the following diﬀerential equations. 1. ( 10 marks) y 00 - y 0 = xe 2 x . 2. ( 10 marks) y 00 + 4 y = sin 2 x. Problem 2 ( 10 marks) . Solve the diﬀerential equation x 2 y 00 + 5 xy 0 + 4 y = 0 , x > 0 . Problem 3 ( 10 marks) . Find a particular solution for the equation y 00 + 4 y = 3 csc x, using the method of variation of parameters. (you may need to use the identities, sin 2 x = 2 sin x cos x , cos 2 x = 1 - 2 sin 2 x and R csc xdx = ln | csc x - cot x | + c. ) Solution of 1.1: Auxiliary equation is r 2 - r = 0. Thus r 1 = 0, r 2 = 1. Thus y c = c 1 + c 2 e x . To ﬁnd a y p we try y = ( Ax + B ) e 2 x . Then y 0 = e 2 x (2 Ax + 2 B + A ) , and y 00 = e 2 x (4 Ax + 4 B + 4 A ) . Plug these in the equation to get e 2 x (2 AX + (2 B + 3 A )) = xe 2 x . 1

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.

{[ snackBarMessage ]}

### Page1 / 3

math1005test2sol - 40 marks-50 minutes MATH1005A Summer2009...

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

View Full Document
Ask a homework question - tutors are online