$("d"y)/("d"x)$ = (4x + y + 1) ......(i)
Put 4x + y + 1 = t .....(ii)
Differentiating w.r.t. x, we get
$4 + ("d"y)/("d"x) = ("dt")/("d"x)$
$("d"y)/("d"x) = "dt"/("d"x) - 4$ ......(iii)
Substituting (ii) and (iii) in (i), we get
$"dt"/("d"x) - 4$ = t
$"dt"/("d"x)$ = u + 4
$"dt"/("t" + 4)$ = dx
Integrating on both sides, we get
$int "dt"/("t" + 4) = int "d"x$
log |t + 4| = x + c
log |(4x + y + 1) + 4| = x + c
log |4x + y + 5| = x + c ......(iv)
When y = 1, x = 0
log |4(0) + 1 + 5| = x + c
c = log |6|
log |4x + y + 5| = x + log 6 ....[From (iv)]
log |4x + y + 5| log 6 = x
$log|(4x + y + 5)/6|$ = x, which is the required particular solution
