Yes, the first one is correct.
The second should be
Using the initial condition, C=1.
Therefore,
(1 + x^2) (dy/dx) = y with y(0) = 1 the answer i am getting is y = e^(arctan x) is this correct?? also for (dy/dt)(cos t) + y(sin t) = 1 with y(Pi/4) = sqrt(2) the answer i am getting is y = (cos t)(tan t) + (1/2)(sqrt(2)) is this correct?