 Solution

Admissions add to the expected value at rate $\alpha$, while the expected aggregate discharge rate is $\delta \mathbb{E}[X(t)]$. ConsequentlyMultiplying this linear ordinary differential equation by the integrating factor $e^{\delta t}$ givesIntegrating from zero, using $N(0)=0$, gives $e^{\delta t}N(t)=\alpha (e^{\delta t}-1)/\delta$. HenceIt starts at zero, has initial slope $\alpha$, and approaches $\alpha /\delta$ as $t\to \infty$, as the admission and discharge rates balance.