Doing this in three steps to get it to fit within the line length limitation
If the supposed solution to the DE is
C1 Sin[2x]+C2 Cos[x]^2+C3-x Cos[2x]/4-Log[Cos[x]-Sin[x]]/8+Log[Sin[x]+Cos[x]]/8+Sin[2x]Log[Cos[2x]]/8
then the third derivative seems to be
third
which is also
-(Sec[2 x] (-2+8 C1+Log[Cos[2 x]]+Cos[4 x] (8 C1+Log[Cos[2 x]])+2 (-2 C2+x) Sin[4 x]))/2
and four times the first derivative seems to be
fourfirst
which is also
Cos[2 x] (8 C1+Log[Cos[2 x]])+2 (-2 C2+x) Sin[2 x]
and the sum of those seems to be
sum
which is also
Sec[2 x]
Can you explain exactly where in these steps you seem to see an error and what the error seems to be?
