Let y(t) be the reserves of oil left in an oil pool at time...

Let $y(t)$ be the reserves of oil left in an oil pool at time $t$. Suppose that extraction reduces reserves at a constant rate equal to $\alpha$. (The rate of decline of reserves is $\alpha$.) If initial reserves at $t=0$ were 500 million barrels, solve for the expression showing reserves as a function of time.

Key Concepts

Rate of Change

Rate of change describes how a quantity varies over time. In mathematical modeling, this concept is represented by a derivative that quantifies the speed at which a quantity is increasing or decreasing. In the context of differential equations, it identifies how rapidly the dependent variable, such as a resource's amount, is changing with respect to time.

Initial Value Problem

An initial value problem involves a differential equation along with a specified value of the unknown function at a given point, usually at the beginning of the interval. This initial condition allows for the determination of a unique solution that fits a real-world scenario or model.

Linear Differential Equation

A linear differential equation is one in which the unknown function and its derivatives appear to the first power and are not multiplied together. Such equations are often easier to solve and provide direct insight into processes where the rate of change is proportional to a constant or the function itself.

Integration

Integration is a core technique in calculus used to find functions when their rates of change are known. In solving differential equations, integration is applied to move from the derivative (rate of change) to the original function, thereby quantifying the cumulative effect over time.

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