If we have a differential equation $\frac{\mathrm{d} y}{\mathrm{d} x} + P(x)y = Q(x)$
where $P(x)$ and $Q(x)$ are functions in ‘$x$’.
Then solution of this differential equation is given by :-
$y(I.F.) = \int Q(x)(I.F.)\;dx + c$
where I.F. is integrating factor which is defined as $e^{\int P(x)\;dx}$ and ‘c‘ is arbitrary constant.
So, here, $P(x)=4$ and $Q(x)=5$
Now, I.F. = $e^{\int 4dx} = e^{4x}$
So, Solution of given differential equation is :-
$ye^{4x} = \int 5e^{4x}\;dx + c$
$\Rightarrow$ $ye^{4x} = \frac{5}{4}e^{4x}\; + c$
$\Rightarrow$ $y = \frac{5}{4}\; + ce^{-4x}$
Now, It is given that $y(0)=2.25$
So, $2.25=\frac{5}{4}+c*e^{-0}$
$\Rightarrow$ $c=1.00$
So, Solution of given differential equation will be :-
$y = \frac{5}{4}\; + 1*e^{-4x}$
$\Rightarrow$ $y = 1.25\; + e^{-4x}$
So, Answer is $(B)$
Reference :- https://en.wikipedia.org/wiki/Integrating_factor