Answer Happy

The differentialequationx2d2ydx2−7xdydx+16y=0has x4 as a solution.Applying reduction order we set y2=ux4.Then (using the prime notation for the derivatives)y2′= y2′′= So, plugging y2 into the left side of the differentialequation, and reducing, we get
x2y2′′−7xy2′+16y2=
The differential equation x2dx2d2y​−7xdxdy​+16y=0 has x4 as a solution. Applying reduction order we set y2​=ux4. Then (using the prime notation for the derivatives) y2′​=y2′′​=​ So, plugging y2​ into the left side of the differential equation, and reducing, we get x2y2′′​−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