Question

Find the differential equation of the family of curves $y=A{e}^{2x}+B{e}^{-2x}$, for different values of $A$ and $B$.

Answer 1

Given,

$y=A{e}^{2x}+B{e}^{-2x}$

differentiating on both sides, we get,

$y^{\prime }=3A{e}^{3x}-2B{e}^{-2x}$

again differentiating on both sides, we get,

$y^{\prime \prime }=9A{e}^{3x}+4B{e}^{-2x}$

Now,

$y^{\prime \prime }-y^{\prime }=9A{e}^{3x}+4B{e}^{-2x}-(3A{e}^{3x}-2B{e}^{-2x})$

$=9A{e}^{3x}+4B{e}^{-2x}-3A{e}^{3x}+2B{e}^{-2x}$

$=6A{e}^{3x}+6B{e}^{-2x}$

$=6(A{e}^{2x}+B{e}^{-2x})$

$=6y$

$\therefore y^{\prime \prime }-y^{\prime }=6y$

Is the required differential equation of family of curves.