Note that the general solution to the homogenous differentia

Note that the general solution to the homogenous differential equation 9 y\" - 8y\' - y = 0 is yh(t) = C1 middot e^-t/9 + C2 middot e^t. (The roots of the characteristic equation are 1 and -1/9) Solve the following initial value problem. 9y\" -8y\' -y= 10e^t y(0) = -4 y\'(0) = -3 Solve the following differential equation. (You will only be able to find the general solution.) 9y\" - 8y\' - y = t^2

Solution

a)general solution is given.

   for particular solution:

    using inverse d operator method,

     y_p = (10/ (9D^2 - 8D - 1)) e^t

   now put D = 1, since for e^at, we put D = a. here a=1

     y_p = (10/ 9x 1^2 -8x1-1) = infinity

so y_p = does not exist.

    y= c1. e^(-t/9) + c2 . e^t

    y\' = (-c1/9) e^(-t/9) + c2 . e^t

at y(0) = -4, -4 = c1 + c2

at y\'(0) = -3, -3 = -c1/9 + c2 => -27 = -c1 + 9c2 => c2 = -31/10

c1 = -9/10

y= -9/10. e^(-t/9) -31/10 . e^t

b) same solution as part (a).