Answer
See below.
Work Step by Step
Take the derivatives of the function $y$. Remember to use chain rule. $$y'(x)=-5c_1e^{-5x}+5c_2e^{5x}$$ $$y''(x)=25c_1e^{-5x}+25c_2e^{5x}$$ Substitute these equations into the differential equation to get $$y''-25y=0$$ $$(25c_1e^{-5x}+25c_2e^{5x})-25(c_1e^{-5x}+c_2e^{5x})=0$$ This satisfies the differential equation, so it must be a solution.
