Solving second order differential equations in quantum mechanics by order reduction. (Q1809890)

Using the technique of order reduction for simplifying the mathematical structure, the author shows that the differential equation of second order \[ (a_2 x^2+ a_1(x)+ a_0)y''+ (b_1x+ b_0) y'+ c_0y= 0 \] has the particular solution \[ y(x)= \gamma_i{d^i\over dx^i}\exp\Biggl[\int {(2a_2 i+ 2a_2- b_1) x+ a_1 i+ a_1- b_0\over a_2 x^2+ a_1 x+ a_0}\,dx\Biggr], \] where \(i\) is a nonnegative integer determined by the condition which is called the condition of order reduction, that is \[ d_0=- (i+ 1)(a_2 i+ 2a_2- b_1). \] The author shows that the approach developed here is easier than the method of generating functions and more powerful than the Frobenius method of power series.