Represent the following families of curves by forming the corresponding differential equations (a, b being parameters): (i) `(x^2)/(a^2)-(y^2)/(b^2)=1` (ii) `y=e^(a x)`

Explanation

(i) ` (x^2)/(a^2)-(y^2)/(b^2)=1ldots `(1)

Differentiating it with respect to `x`, we get

`{2x}/a^2-{2y}/b^2{dy}/{dx}=0ldots`(2)

Differentiating it with respect to `x`, we get

`2/a^2-2/b^2({dy}/{dx})^2-{2y}/b^2{d^2y}/{dx^2}=0`

`implies {b^2}/{a^2}=y{d^2y}/{dx^2}+({dy}/{dx})^2ldots`(3)

`implies {b^2}/{a^2}=y/x{dy}/{dx}ldots`(4) (Using (2))

Eliminating `a` and `b` from (3) and (4), we get
...