Exponential Growth and Decay Differential Equation

dy/dt = ky

This problem introduces one of the most fundamental concepts in differential equations: exponential growth and decay. The provided equation is a simple differential equation that models a wide variety of real-world scenarios, such as population growth, radioactive decay, and interest calculations in financial models. The equation is defined by $\frac{dy}{dt} = ky$, where the solution involves exponential functions based on the initial conditions and the constant $k$.

Understanding this equation involves recognizing it as a separable differential equation, where variables can be rearranged to integrate both sides separately. This leads to a solution that is an exponential function. Key concepts include recognizing the role of $k$, which determines the growth or decay rate, and interpreting initial conditions to solve for particular solutions.

In tackling problems like these, students learn essential techniques not only for solving differential equations but also for appreciating their practical applications in modeling dynamic systems. Identifying the differential equation in context and applying integration techniques are crucial skills developed through problems of this nature.