$y(x)=e^{ax}(c_{1}cosbx+c_{2}sinbx),y^{\text{'}\text{'}}-2a{y}^{\text{'}}+(a^2+b^2)y=0,$ where a and $b$ are constants.

For Problems $7-21,$ verify that the given function is a solution to the given differential equation $(c_1$ and $c_2$ are arbitrary constants$),$ and state the maximum interval over which the solution is valid.