For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

xy = a e^x + b e^–x + x² :

It is given that xy = a e^x + b e^–x + x²

Now, differentiating both sides w.r.t. x, we get,

Now, Again differentiating both sides w.r.t. x, we get,

Now, Substituting the values of ’ and in the given differential equations, we get,

LHS =

= x(ae^x +be^-x + 2) + 2(ae^x - be^-x + 2) –x(ae^x +be^-x + x²) + x² – 2

= (axe^x +bxe^-x + 2x) + 2(ae^x - be^-x + 2) –x(ae^x +be^-x + x²) + x² – 2

= 2ae^x -2be^-x +x² + 6x -2

≠ 0

⇒ LHS ≠ RHS.

Therefore, the given function is not the solution of the corresponding differential equation.