Let `f:R to R` be a differentiable function with `f(0)=0`. If `y=f(x)` satisfies the differential equation
`(dy)/(dx)=(2+5y)(5y-2)`, then the value of `lim_(x to oo) f(x)` is……………….

Answer

D

Explanation

`(dy)/(dx) = (5y+2)(5y-2)`
`rArr 1/25int(dy)/(y^(2)-(2/5)^(2))=intdx`
`=1/25.5/4log_(e)|(y-2)/(y+2/5)|=x+c`
`=1/20 log_(e)|(5y-2)/(5y+2)|=x+c`
Given that `f(0)=y(0)=0`.
`therefore c=0`
Hence, `|(2-5y)/(2+5y)|=e^(20x)`
`therefore lim_(x to infty) |(2-5y)/(2+5y)|=lim_(x to -infty)e^(20x)`
`rArr lim_(x to -infty)|(2-5y)/(2+5y)|=0`
`lim_(x to -infty)y=2/5=0.4`