The differential equaiotn which represents the family of curves `y=C_(1)e^(C_(2)x)`, where `C_(1)` and `C_(2)` are arbitrary constants, is

Options

A. `y^(')=y^(2)`
B. `y^('')=y^(')y`
C. `yy^('')=y^(')`
D. `yy^('')=(y^('))^(2)`

Answer

D

Explanation

`y=c_(1)e^(c_(2)x)`………..(i)
`therefore y^(')=c_(2)c_(1)e^(c_(2)x)=c_(2)y`…………..(ii)
`therefore y^('')=c_(2)y^(')`………..(iii)
From (ii), we get
`c(2)=y^(')/y`
From (ii) and (iii)
We get `y^('')/y^(')=y^(')/y` or `yy^('')=y^('^(2))`