Form the differential equation of the following family of curves:
x y = A e^x + B e^–x + x²

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Solution

The equation of the family of curves is
x y = A e^x + B e^–x + x^{2 ...(1)}
Differentiating both sides w.r.t. x, we get,

straight x dy over dx plus straight y space.1 space equals space Ae to the power of straight x minus Be to the power of negative straight x end exponent plus 2 straight x

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

straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus dy over dx.1 plus dy over dx space equals space straight A space straight e to the power of straight x plus Be to the power of negative straight x end exponent plus 2

straight x fraction numerator straight d squared straight y over denominator dx squared end fraction plus 2 dy over dx space equals space xy space minus straight x squared plus 2

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

which is required differential equation.

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