The differential equation x2 dx2 d²y dy 7x- +16y=0 dx has x4 as a solution. Applying reduction order we set y2 = ux¹. Then (using the prime notation for the derivatives) y2 = y₁₂ = So, plugging y2 into the left side of the differential equation, and reducing, we get x²y 7xy2+16y2 The reduced form has a common factor of x5 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 = Finally y2 = and the general solution is

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