brings me this answer:
So the circled with red it's incorrect in my opinion
Probably the first mistake causes the second.
What do you think?
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
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
Cos[2 x] (8 C1+Log[Cos[2 x]])+2 (-2 C2+x) Sin[2 x]
and the sum of those seems to be
Can you explain exactly where in these steps you seem to see an error and what the error seems to be?
And my method brings me this answer:
and here is his comprobation too: http://www.wolframalpha.com/input/?i=(cos2x(1-(x%2F4))%2Bsin2x(1%2B(ln(cos2x)%2F8))%2B((ln(sec2x%2Btan2x))%2F8))%27%27%27%2B4(cos2x(1-(x%2F4))%2Bsin2x(1%2B(ln(cos2x)%2F8))%2B((ln(sec2x%2Btan2x))%2F8))%27%3Dsec2x