{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# sf - Math 5525 Final Exam Problems and Solutions Problem...

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

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 0 ( 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 0 + 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 0 = ln x + c, w = x ln x - x + cx + c 1 . Therefore, the general solution is y = ( c 1 + c 2 x + x ln x ) e - x .

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 ]}

### 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