{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

4. ODE_nonconstant_coefficients

4. ODE_nonconstant_coefficients - 2nd order linear...

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

View Full Document Right Arrow Icon
2 nd order, linear, homogeneous, non-constant coefficient ODEs Example: Cauchy-Euler (Euler) O.D.E. 0 2 2 2 = + + cy dx dy bx dx y d ax here P(x)= 2 ax , Q(x)=bx, R(x)=c and a, b, c = constants This is one of the few cases where an exact solution can be found without using infinite series. Introduce a change of independent variable. Let t = ln x ( or x=e t ). Use the chain rule to change the ODE for y(x) into an ODE for y(t): dt t dy x dt t y d x dx x y d dt t dy x dx dt dt t dy dx x dy ) ( 1 ) ( 1 ) ( ) ( 1 ) ( ) ( 2 2 2 2 2 2 = = = Then, with this substitution, the ODE above reduces to 0 ) ( ) ( ) ( ) ( 2 2 = + + t cy dt t dy a b dt t y d a This constant coefficient homogeneous ODE can be solved easily for y(t) , and then we substitute t = ln x to find the y(x) solution. For applications to be discussed later in the course, the y(x) solution can be solved easily for y(t) as in section 1.2, and then we substitute t = ln x to find the y(x) solution that we want. For the applications we will see later, the y(x) solution usually is of the form y(x) = C 1 x p + C 2 x –p , where p is a constant, and C 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
Image of page 2
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