Answer
See below.
Work Step by Step
Take the derivatives of the function. Remember to use chain rule. $$y(x)=c_1cosh(3x)+c_2sinh(3x)$$ $$y'(x)=3c_1sinh(3x)+3c_2cosh(3x)$$ $$y''(x)=9c_1cosh(3x)+9c_2cosh(3x)$$ Substituting these functions into the differential equation yields $$y''-9y=0$$ $$(9c_1cosh(3x)+9c_2cosh(3x))-9(c_1cosh(3x)+c_2sinh(3x))=0$$ $$0=0$$ This means that the solution is valid for all real values. The interval linked to this solution is $(-\infty,\infty)$.
