The differential equation which represents the family of curves `y=c_(1)e^(c_(2^(x)` where `c_(1)andc_(2)` are arbitary constants is

Options

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

Answer

D

Explanation

The equation of the family of curves is
`y=C_(1)e^(C_(2)x)rArr log y =log C_(1)+C_(2)x`
Differentiating w.r. to x, we get
`(1)/(y)(dy)/(dx)=C_(2)`
Differentiating w.r. to x, we get
`(1)/(y)(d^(2)y)/(dx^(2))-(1)/(y^(2))((dy)/(dx))^(2)=0rArry(d^(2)y)/(dx^(2))=((dy)/(dx))^(2)or, yy''=(y')^(2)`