Solving and Checking Solutions

Mathematica can take some of the drudgery out of solving differential
equations.  For instance, suppose we want to solve y' = 2y + sin t
subject to the initial condition y(π) = 1.  The appropriate integrating
factor for this differential equation is

mu[t_]= Exp[(∫_Pi^t)-2 \[DifferentialD]s]

We then compute that the solution is

y [t_] = Simplify[(1+(∫_Pi^t)mu[s] Sin[s] \[DifferentialD]s )/mu[t]]

Of course, you would never do something silly like lose a minus sign or inadverdantly drop a "2" when computing the solution to an ODE.
On the other hand, you might be a bit suspicious of Mathematica. 
Fortunately, you always have the option of making Mathematica check
its solutions:

Simplify[(∂_t) y[t]  -  2 y[t]  -  Sin[t]]