math-4b-cauchy-euler34764e76c2e465cc8cb6ff0000369d41.pptx

math-4b-cauchy-euler34764e76c2e465cc8cb6ff0000369d41.pptx -...

This preview shows page 1 - 5 out of 17 pages.

Differential Equations Cauchy-Euler Equations Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Image of page 1

Subscribe to view the full document.

A Cauchy-Euler equation is a specific type of D.E. that can be put in the form: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB To solve this we will make this variable substitution: This one is 2 nd -order, but the solution method we will see generalizes to higher order equations as well. In the original equation, y is a function of t. When we substitue, we are thinking of y as a function of x, which in turn is a function of t. So it is very much like solving an integral by substitution, where you choose an intermediate variable that simplifies the integrand. It will help if we use the Leibniz notation for our derivatives. Our equation is: ) t ( g y y t y t 2 x e t ) t ln( x ) t ( g y dt dy t dt y d t 2 2 2
Image of page 2
Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB We need to find the derivatives with the chain rule, since we now have a function y(x(t)): We are using this substitution: t 1 dt dx ) t ln( x ) t ( g y dt dy t dt y d t 2 2 2 t 1 dx dy dt dx dx dy dt dy 2 2 2 2 2 2 2 2 2 2 2 2 2 t 1 dx dy t 1 dx y d dt y d t 1 dx dy t 1 dt dx dx y d dt y d t 1 dt d dx dy t 1 dx dy dt d t 1 dx dy dt d dt y d
Image of page 3

Subscribe to view the full document.

Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Next, plug in our expressions to get the equation in terms of x rather than t:
Image of page 4
Image of page 5

{[ 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