1550078481-Ordinary_Differential_Equations__Roberts_

256 Ordinary Differential Equations

11. y" -y' - 2y = O; y(O) = 0, y'(O) = 3 (HINT: See exercise 2.)

11 1

12. y" - y' - 2y = x^2 ; y(O) = 4' y'(O) = 2


13. y" - 2y' + y = 2 sinx; y(O) = -2, y' (0) = 0


14. y"' - y" + 4y' - 4y = O; y(O) = 0, y' (0) = 5 y" (0) = 5
    (HINT: See exercise 5.)


15. Show that for any continuous function f(x) and any constant a =I-0,
    the solution of the initial value problem

y" + a^2 y = f(x); y(O) = 0, y'(O) = 0

is

y(x) = {~sin ax}* f(x) =~lax sina(x - ~)!(~) d~

= ~ { sin ax lax f(O cos a~ d~ -cos ax lax f(~) sin a~ d~}.

16. Show that for any continuous function f(x) and any constant a =f. 0,

the solution of the initial value problem

y" - a^2 y = j(x); y(O) = 0, y'(O) = 0

is

y(x) = { 2~ eax} * f(x) - { 21a e-ax} * j(x)

17. Show t hat for a ny continuous function j(x) and any constant a =f. 0,

the solution of the initial value problem

y" - 2ay' + a^2 y = f(x ); y(O) = 0, y' (0) = 0

is

y(x) = {xeax} * f(x) =lax (x - ~)ea(x-0 f(~) d~