" (i) "xy=ae^(x)+be^(-x)+x^(2)quad :quad x(d^(2)y)/(dx^(2))+2(dy)/(dx)-xy+x^(2)-2=0

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation. xy = ae^(x) + be^(-x) + x^(2) : x (d^(2)y)/(dx^(2)) + 2(dy)/(dx) - xy + x^(2) - 2 = 0

If y=ae^(2x)+be^(-x), show that (d^(2)y)/(dx^(2))-(dy)/(dx)-2y=0

(x^(2)+xy)(dy)/(dx)=x^(2)+y^(2)

If y=x^(x) ,then xy (d^(2)y)/(dx^(2))-x((dy)/(dx) )^(2)=

x^(2)(dy)/(dx)+y^(2)=xy

x^(2)(dy)/(dx)=x^(2)+xy+y^(2)

((dy)/(dx))^(2)-(x+y)(dy)/(dx)+xy=0