lecture_13 (dragged) 2

lecture_13 (dragged) 2 - MA 36600 LECTURE NOTES MONDAY...

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

View Full Document Right Arrow Icon
MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16 3 If the characteristic equation a r 2 + b r + c = 0 has two distinct real roots r 1 and r 2 , then the general solution to the di ff erential equation is y ( t ) = c 1 e r 1 t + c 2 e r 2 t for some constants c 1 and c 2 . Recall that a quadratic equation a r 2 + b r + c = 0 has two distinct real roots if its discriminant is positive: b 2 4 a c > 0. Example. Say that we wish to find the general solution to the di ff erential equation y + 5 y + 6 y = 0 . To solve this problem, we consider the characteristic equation. In this case, it is r 2 + 5 r + 6 = 0 . The polynomial factors as r 2 + 5 r + 6 = ( r + 2) ( r + 3) = r 1 = 2 , r 2 = 3 . Since these two roots are distinct, the general solution to the di ff erential equation is y ( t ) = c 1 e 2 t + c 2 e 3 t for some constants c 1 and c 2 . Proof of the General Solution. We return to the claim made earlier about the general solution to the homogeneous linear second order di ff erential equation with constant coe cients. In fact, we make a stronger claim. Consider the initial value problem
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