Answer
$y=-2t-4+5 e^{1/2t}$
Work Step by Step
We are given that the differential equation : $ 2\dfrac{dy}{dt}-y=2t$
or, $\dfrac{dy}{dt}-\dfrac{y}{2}=t$
and the general solution can be written as:
$y=e^{-(-1/2)t} \int te^{-1/2t} \ dt=e^{1/2t} \int te^{-1/2t}\ dt $
Integrate to obtain:
$y=e^{1/2t}(-2t e^{-1/2t}-4e^{-1/2t}+C)$
This implies that $y=-2t-4+C e^{1/2t}$
After applying the initial conditions, $y=1$ when $t=0$, we get $C=5$
Therefore, we have: $y=-2t-4+5 e^{1/2t}$
