Reduction of order (or d’Alembert reduction) is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y_1(x) is known and a second linearly independent solution y_2(x) is desired. The method also applies to n-th order equations. In this case the ansatz will yield an (n−1)-th order equation for v.

Second-order linear ordinary differential equations

An example

Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE) where a, b, c are real non-zero coefficients. Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, b^2 - 4 a c, vanishes. In this case, from which only one solution, can be found using its characteristic equation. The method of reduction of order is used to obtain a second linearly independent solution to this differential equation using our one known solution. To find a second solution we take as a guess where v(x) is an unknown function to be determined. Since y_2(x) must satisfy the original ODE, we substitute it back in to get Rearranging this equation in terms of the derivatives of v(x) we get Since we know that y_1(x) is a solution to the original problem, the coefficient of the last term is equal to zero. Furthermore, substituting y_1(x) into the second term's coefficient yields (for that coefficient) Therefore, we are left with Since a is assumed non-zero and y_1(x) is an exponential function (and thus always non-zero), we have v'' = 0. This can be integrated twice to yield where c_1, c_2 are constants of integration. We now can write our second solution as Since the second term in y_2(x) is a scalar multiple of the first solution (and thus linearly dependent) we can drop that term, yielding a final solution of Finally, we can prove that the second solution y_2(x) found via this method is linearly independent of the first solution by calculating the Wronskian Thus y_2(x) is the second linearly independent solution we were looking for.

General method

Given the general non-homogeneous linear differential equation and a single solution y_1(t) of the homogeneous equation [r(t)=0], let us try a solution of the full non-homogeneous equation in the form: where v(t) is an arbitrary function. Thus and If these are substituted for y, y', and y'' in the differential equation, then Since y_1(t) is a solution of the original homogeneous differential equation,, so we can reduce to which is a first-order differential equation for v'(t) (reduction of order). Divide by y_1(t), obtaining One integrating factor is given by, and because this integrating factor can be more neatly expressed as Multiplying the differential equation by the integrating factor \mu(t), the equation for v(t) can be reduced to After integrating the last equation, v'(t) is found, containing one constant of integration. Then, integrate v'(t) to find the full solution of the original non-homogeneous second-order equation, exhibiting two constants of integration as it should: