sf - Math 5525. May 11, 2010. Final Exam. Problems and...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 5525. May 11, 2010. Final Exam. Problems and Solutions. Problem 1 (10 points) Find the general solution of the problem dy dx = y 2- x 2 xy + x y . Solution. dy dx = y 2- x 2 xy + x y = y x , v = y x = v + x dv dx = v, dv dx = 0 , v = C, y = Cx. Problem 2 (15 points) Find the general solution of the equation y x dx + ( y 3 + ln x ) dy = 0 . Solution. This equation has the form Mdx + Ndy = 0 with M y = N x = 1 /x , i.e. it is an exact equation. Therefore, the general solution has the form u ( x,y ) = C , where u x = M, u y = N . u x = M = y x = u = ( x,y ) = y ln x + g ( y ); u y = ln x + g ( y ) = N = y 3 +ln x = g ( y ) = y 4 4 . The general solution is 4 u = 4 y ln x + y 4 = C 1 . Problem 3 (20 points) Find the general solution of the differential equation y 00 + 2 y + y = 1 xe x . Solution. We rewrite the given equation as follows Ly = ( D + 1) 2 y = e- x 1 x . One can find a particular solution in the form y p = e- x w ( x ). We have ( D + 1) 2 y p = ( D + 1) 2 ( e- x w ) = e- x w 00 = e- x 1 x w 00 = 1 x , w = ln x + c, w = x ln x- x + cx + c 1 ....
View Full Document

Page1 / 3

sf - Math 5525. May 11, 2010. Final Exam. Problems and...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online