the differential equation has as a solution. applying reduction of order we set . then (using the prime notation for the derivatives) so, substituting and its derivatives into the left side of the differential equation, and reducing, we get the reduced form has a common factor of which we can divide out of the equation. since this equation does not have any terms in it we can make the substitution giving us the first order linear equation in : . if we use c as the constant of integration, the solution to this equation is integrating to get , and then finding gives the general solution: