The differential equation has as a solution. x^2 d^2y/dx^2- 7x dy/dx + 16y = 0
Applying reduction order we set y^2 = ux^4. Then (using the prime notation for the derivatives). So, plugging y^2 into the left side of the differential equation, and reducing, we get
x^2y"^2 - 7xy'^2 + 16y^2 =
The reduced form has a common factor of x^5 which we can divide out of the equation so that we have xu" + u' = 0. Since this equation does not have any u terms in it we can make the substitution w = u' giving us the first order linear equation xw' + w = 0. This equation has integrating factor for x > 0. If we use a as the constant of integration, the solution to this equation is w =. Integrating to get u, and using b as our second constant of integration we have u = 0. Finally y^2 =and the general solution is.