Answer
The differential equation is satisfied.
Work Step by Step
As we are given that $y=\int_{0}^{x} (1+2 \sqrt {\sec t}) dt$
Now, $y'=1+2 \sqrt {\sec x}$
and $y''=2 (\dfrac{1}{2}) (\sec x)^{-(1/2)} (\sec x\tan x)=(\sec x)^{-\frac{1}{2}+1} \tan x$
$\implies y''=(\sec x)^{\frac{1}{2}} \tan x=\sqrt {\sec x} \tan x$
Apply initial conditions $y(0)=0$
$\implies y'(0)=1+2 \sqrt {\sec (0)}=1+2(1)=3$
Hence, the differential equation is satisfied.
