replaced by The particular solution to the non-homogeneous equation can then be written as Intuitive explanation Consider the equation of the forced dispersionless spring, in suitable units: Here is the displacement of the spring from the equilibrium , and is an external applied force that depends on time. When the external force is zero, this is the homogeneous equation (whose solutions are linear combinations of sines and cosines, corresponding to the spring oscillating with constant total energy). We can construct the solution physically, as follows. Between times and , the momentum corresponding to the solution has a net change (see: Impulse (physics)). A solution to the inhomogeneous equation, at the present time , is obtained by linearly superposing the solutions obtained in this manner, for going between 0 and . The homogeneous initial-value problem, representing a small impulse being added to the solution at time , is The unique solution to this problem is easily seen to be . The linear superposition of all of these solutions is given by the integral: To verify that this satisfies the required equation: as required (see: Leibniz integral rule). The general method of variation of parameters allows for solving an inhomogeneous linear equation by means of considering the second-order linear differential operator L to be the net force, thus the total impulse imparted to a solution between time s and s+ds is F(s)ds. Denote by the solution of the homogeneous initial value problem Then a particular solution of the inhomogeneous equation is the result of linearly superposing the infinitesimal homogeneous solutions. There are generalizations to higher order linear differential operators. In practice, variation of parameters usually involves the fundamental solution of the homogeneous problem, the infinitesimal solutions then being given in terms of explicit linear combinations of linearly independent fundamental solutions. In the case of the forced dispersionless spring, the kernel is the associated decomposition into fundamental solutions. Examples First-order equation The complementary solution to our original (inhomogeneous) equation is the general solution of the corresponding homogeneous equation (written below): This homogeneous differential equation can be solved by different methods, for example separation of variables: The complementary solution to our original equation is therefore: Now we return to solving the non-homogeneous equation: Using the method variation of parameters, the particular solution is formed by multiplying the complementary solution by an unknown function C(x): By substituting the particular solution into the non-homogeneous equation, we