1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: The Principle of Superposition for Homogeneous Equations (DiffEq)

  1. Oct 5, 2014 #1
    1. The problem statement, all variables and given/known data
    Verify that [itex]e^x[/itex] and [itex]e^-x[/itex] and any linear combination [itex]c_1e^x + c_2e^{-x} [/itex] are all solutions of the differential equation:
    [itex] y'' - y = 0 [/itex]
    Show that the hyperbolic sine and cosine functions, sinhx and coshx are also solutions

    2. Relevant equations
    Principle of Superposition for Homogeneous Equations
    [itex] y'' + p(x)y' + q(x)y = 0 [/itex]
    [itex] y(x) = c_1y_1(x) = c_2y_2(x) [/itex]

    3. The attempt at a solution
    I am not having any trouble on the first part, here is my solution:
    [itex] y_1(x) = e^x [/itex]
    [itex] y_2(x) = e^{-x} [/itex]
    [itex] y'' - y = 0 [/itex]
    [itex] y = c_1e^x + c_2e^{-x} [/itex]
    [itex] y' = c_1e^x - c_2e^{-x} [/itex]
    [itex] y'' = c_1e^x + c_2e^{-x} [/itex]
    [itex] (c_1e^x + c_2e^{-x}) - (c_1e^x + c_2e^{-x}) = 0 [/itex]
    [itex] 0 = 0 [/itex]
    Now, on the second part of the problem I run into problems, here is what I have so far:
    [itex] y_1(x) = sinh(x) [/itex]
    [itex] y_2(x) = cosh(x) [/itex]
    [itex] y = c_1cosh(x) + c_2sinh(x) [/itex]
    [itex] y' = -c_1sinh(x) + c_2cosh(x) [/itex]
    [itex] y'' = -c_1cosh(x) - c_2sinh(x) [/itex]
    [itex] (-c_1cosh(x) - c_2sinh(x)) - (c_1cosh(x) + c_2sinh(x)) = 0 [/itex]
    However, the last equation is not true and I am not sure where I went wrong....
    Last edited: Oct 5, 2014
  2. jcsd
  3. Oct 5, 2014 #2


    Staff: Mentor

    Your derivative for cosh(x) is wrong. d/dx(cosh(x)) = sinh(x).
  4. Oct 5, 2014 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The derivatives of hyperbolic sinh cosh functions don't have a minus sign like the ordinary sines and cosines do.
  5. Oct 5, 2014 #4
    thank you both...I was so confused for a moment XD
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted