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11.4: Ordinary Homogeneous Second-order Differential Equations
https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Tatum)/11%3A_Simple_and_Damped_Oscillatory_Motion/11.04%3A_Ordinary_Homogeneous_Second-order_Differential_Equations
the highest derivative is a second derivative - and therefore there can be only two arbitrary constants of integration in the solution - and we already have two in Equation $\ref{11.4.1}$ , and cons...the highest derivative is a second derivative - and therefore there can be only two arbitrary constants of integration in the solution - and we already have two in Equation $\ref{11.4.1}$ , and consequently there are no further solutions. It will not take long to discover that solutions of the form $y=e^{kx}$ satisfy the Equation, because then $y^{'}=ky$ and $y^{''}=k^{2}y$ , and, if you substitute these in Equation $\ref{11.4.1}$ , you obtain

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