Find the differential equation of the family of curves y = Ae^2x + Be^−2x, where A and B are arbitrary constants.

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Solution

The equation of the family of curves is
$y = A e^{2 x} + B e^{- 2 x}$ ...(1)
where $A and B$ are arbitrary constants.
This equation contains two arbitrary constants, so we shall get a differential equation of second order.
Differentiating equation (1) with respect to x, we get
$\frac{d y}{d x} = 2 A e^{2 x} - 2 B e^{- 2 x}$ ...(2)
Differentiating equation (2) with respect to x, we get
$\frac{d^{2} y}{d x^{2}} = 4 A e^{2 x} + 4 B e^{- 2 x} \Rightarrow \frac{d^{2} y}{d x^{2}} = 4 (A e^{2 x} + B e^{- 2 x}) \Rightarrow \frac{d^{2} y}{d x^{2}} = 4 y It is the required differential equation .$

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