I was solving these DEs

$\frac{dx}{dt} = -y+t$  

$\frac{dy}{dt} = x-t$

Doing it by elimination they become

$x''+x = t+1$  

$y''+y = t-1$

...for $x$ and $y$ respectively. If I solve those I get

$x = c\_1\cos(t) + c\_2\sin(t) + t + 1$  

$y = c\_1\cos(t) + c\_2\sin(t) + t - 1$

Now, it makes sense to me that the general solution (Cos and Sin part) is the same for both cases, but the solution says

$x = c\_1\cos(t) + c\_2\sin(t) + t + 1$  

$y = c\_1\sin(t) - c\_2\cos(t) + t - 1$

And this happens with the solutions of a few other problems too, so I'm not quite convinced it's just a print error. Am I doing something wrong?