This paper proposes a differential equation inventory model that incorporates partial backlogging and deterioration. Holding cost and demand rate are time dependent. Shortages are allowed and assumed to be partially backlogged. Two versions are presented, the first one with deterministic values of the parameters and the second one taking into the account the interval uncertainty of the parameters. In the crisp case, Taylor’s series expansion is used, and graphically shown that the cost function is convex. While, in the case of intervals, the interval arithmetic is used and then the problem is transformed into a multi-objective non-linear optimization problem and an interval objective function. To solve this problem, the weighted-sum method is used. The proposed procedure is validated with the help of a numerical example. Sensitivity analysis on various parameters has also been carried out.