Question

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

Answer 1

The equation of the family of curves is \[y = A e^{2x} + B e^{- 2x}\]                                     ...(1)
where \[A\text{ and }B\] 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

\[\frac{dy}{dx} = 2A e^{2x} - 2B e^{- 2x}\]                               ...(2)
Differentiating equation (2) with respect to x, we get
\[\frac{d^2 y}{d x^2} = 4A e^{2x} + 4B e^{- 2x} \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 4\left( A e^{2x} + B e^{- 2x} \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 4y\]
It is the required differential equation .