The differential equation has as a solution. Applying reduction order we set y = ux¹. Then (using the prime notation for the derivatives) y' = x² d²y dx² So, plugging y into the left side of the differential equation, and reducing, we get dy 7x + 16y=0 dx x²y" 7xy + 16y= The reduced form has a common factor of 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. 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 the general solution is y = The differential equationhas as a solution.Applying reduction order we set y = ux¹.Then (using the prime notation for the derivatives)y' =x²d²ydx²So, plugging y into the left side of the differential equation, and reducing, we getdy7x + 16y=0dxx²y" 7xy + 16y=The reduced form has a common factor of 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.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 the general solution is y =

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Answered: The differential equation has as a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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