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y=ae^(x)+be^(-x)+x^(2):x(d^(2)y)/(dx^(2)...

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

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

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

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

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

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

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

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

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

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