## Differential Equations and Linear Algebra (4th Edition)

Take derivatives of the equation. Don't forget to use chain rule. $$y(x)=c_1e^x+c_2e^{-x}$$ $$y'(x)=c_1e^x-c_2e^{-x}$$ $$y''(x)=c_1e^x+c_2e^{-x}$$ Substituting these functions into the differential equation yields $$y''-y=0$$ $$c_1e^x+c_2e^{-x}-c_1e^x-c_2e^{-x}=0$$ $$0=0$$ This statement is always true, therefore, this solution is valid always on any interval.