General Solution of a Class of Second Order Linear Differential Equations with Variable Coefficients

A second-order linear differential equation y″ + P(x) y' + Q(x) y = f(x)(1) with variable coefficients was converted into the form of z″ + [2φ'(x) + P(x) ]z' + { [φ'(x) ]2 + φ″(x) + P(x) φ'(x) + Q(x) } z = f(x) e-φ(x) using variable substitution of y = zeφ(x).Then the general solutions of equation(1) and its corresponding homogeneous differential equation were obtained respectively in the light of five relationships according to five relationships between P(x) and Q(x).