Assume thatpandqare continuous onIfunctions. If you know two linearlyindependent solutions to this linear equation then you can write t he general solution astheir linear combination. Therefore you are interested in finding another solution which islinearly independent fromy1. You can accomplish this by using a method that is calledReduction of Order. Seek the second solutiony2in the formy2(x) =v(x)y1(x), where youdo not knowv(x). Substitutey2in the equation, thus obtaining a second order, reducibleequation forv. This equation you can solve by a substitutionz=v′. Findz, then integrateit to findv. Then observe that the expression fory2contains a linear summand ofy1. Theleftover is the second, simple, linear independent ofy1solution of the original