the differential equation has as a solution. applying reduction of order we set . then (using the prime notation for the derivatives) so, plugging 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 so that we have . since this equation does not have any u terms in it we can make the substitution giving us the first order linear equation . this equation has integrating factor for x > 0. if we use a as the constant of integration, the solution to this equation is integrating to get u, and using b as our second constant of integration we have finally and the general solution is