{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

set2 - $ Set 2 First Order ODEs Part 1 Kyle A Gallivan...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
a39 a38 a36 a37 Set 2: First Order ODEs - Part 1 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1
Image of page 1

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

View Full Document Right Arrow Icon
a39 a38 a36 a37 First Order ODE Initial Value Problem t R mapsto→ y ( t ) R ( t, y ) R 2 mapsto→ f ( t, y ) R y ( t ) is the highest order of derivative involved The initial value problem: y = f ( t, y ) , y ( t 0 ) = y 0 no general solution so we consider subclasses: linear equations separable equations exact equations 2
Image of page 2
a39 a38 a36 a37 Undetermined Coefficients The Undetermined Coefficients Method: hypothesizes a form of the solution that is a function of a set of coefficients substitutes the form of the solution into the differential equation solves for the unknown coefficients if possible This will be used later on more complicated problems but we apply it to a simple problem here. 3
Image of page 3

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

View Full Document Right Arrow Icon
a39 a38 a36 a37 Undetermined Coefficients Consider the general first order linear constant-coefficient ODE y = ay b (1) where a negationslash = 0 and b are real constants. if b = 0 we know y 1 = ay 1 y 1 ( t ) = Ce at Equation (1) adds a constant to y 1 to get y Equation (1) says y proportional to y Hypothesize that there is a constant k such that y ( t ) = y 1 ( t ) + k 4
Image of page 4
a39 a38 a36 a37 Undetermined Coefficients substitute y ( t ) = y 1 ( t ) + k into ODE RHS : ay b = ay 1 ( t ) + ak b = aCe at + ak b substitute y ( t ) = y 1 ( t ) + k into ODE LHS : dy dt = d dt ( y 1 ( t ) + k ) = d dt ( Ce at ) + d dt ( k ) = aCe at equate RHS and LHS to get conditions on k and check for consistency with assumptions: if aCe at = aCe at + ak b then 0 = ak b k = b a 5
Image of page 5

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

View Full Document Right Arrow Icon
a39 a38 a36 a37 Undetermined Coefficients Since a negationslash = 0 , k is a well-defined constant.
Image of page 6
Image of page 7
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