To solve a nonhomogeneous linear second-order differential equation, first find the general solution to the complementary equation, then find a particular solution to the nonhomogeneous equation To find a general solution for a homogeneous second-order differential equation, we must find two linearly independent solutions.

If y 1 ( x ) and y 2 ( x ) are linearly independent solutions to a second-order, linear, homogeneous differential equation, then the general solution is given by y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) . The form of the general solution varies depending on whether the characteristic equation has distinct, real roots; a single, repeated real root; or complex conjugate roots

linearly independent — a set of functions f 1 ( x ) , f 2 ( x ) ,…, f n ( x ) f 1 ( x ) , f 2 ( x ) ,…, f n ( x ) for which there are no constants c 1 , c 2 ,… c n , such that c 1 f 1 ( x ) + c 2 f 2 (…. boundary conditions — the conditions that give the state of a system at different times, such as the position of a spring-mass system at two different times. characteristic equation — the equation a λ 2 + b λ + c = 0 for the differential equation a y ″ + b y ′ + c y = 0. particular solution — a solution y p ( x ) of a differential equation that contains no arbitrary constants.