The solution of the differential equation `(dy)/(dx)=(4x+y+1)^(2)`, is

Options

A. `(4x+y+1)=tan(2x+c)`
B. `(4+y+1)^(2)=2tan(2x+c)`
C. `(4x+y+1)^(3)=3tan(2x+c)`
D. `(4x+y+1)=2tan(2x+c)`

Answer

D

Explanation

Given, `(dy)/(dx)=(4x+y+1)^(2)`
Put 4x+y+1=v
`implies(dy)/(dx)=(dv)/(dx)-4`
`therefore(dv)/(dx)-4=v^(2)implies(dv)/(v^(2)+4)=dx`
`implies(1)/(2)tan^(-1)((v)/(2))=x+c` [integrating]
`impliestan^(-1)((4x+y+1)/(2))=2x+c`
`implies4x+y+1=2tan(2x+c)`