{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Differential Equations Solutions 114

Differential Equations Solutions 114 - 124 Chapter 20...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
124 Chapter 20. Solutions: Solution of Ordinary Differential Equations CHALLENGE 20.15. Using the notation of the pointer, we let a ( t ) = 1 > 0, b ( t ) = 8 . 125 π cot((1 + t ) π/ 8), c ( t ) = π 2 > 0, and f ( t ) = 3 π 2 . These are all smooth functions on [0 , 1]. Since c ( t ) = π 2 / 2 > 0 and 1 0 [ f ( t )] 2 d t = π 4 4 , the solution exists and is unique. Since f ( t ) < 0, the Maximum Principle tells us that max t [0 , 1] u ( t ) max( 2 . 0761 , 2 . 2929 , 0) = 0 . Letting v ( t ) = 3, we see that v ( t ) + 8 . 125 π cot((1 + t ) π/ 8) v ( t ) + π 2 v ( t ) = 3 π 2 and v (0) = v (1) = 3. Therefore the Monotonicity Theorem says that u ( t ) v ( t ) for t [0 , 1]. Therefore we conclude 3 u ( t ) 0 for t [0 , 1]. Note on how I constructed the problem: The true solution to the prob- lem is u ( t ) = cos((1 + t ) π/ 8)
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern