the differential equation has as a solution. applying reduction order we set . then (using the prime notation for the derivatives) u'x^4 4ux^3 u''x^4 8u'x^3 12ux^2 so, plugging into the left side of the differential equation, and reducing, we get u''x^6 u'x^5 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 x for x > 0. if we use