Solving Exponential Growth Differential Equations

Solve a differential equation of the form $\frac{dy}{dt} = ky$ using separation of variables and integrating both sides to find $y = y_0 e^{kt}$.

Differential equations can often be encountered when dealing with natural growth or decay processes. One common type of differential equation is the exponential growth or decay equation, expressed in the form $\frac{dy}{dt} = ky$, where y is a function of t, and k is a constant. This type of equation describes how the rate of change of a quantity is proportional to the current quantity itself. These equations can be found in numerous fields ranging from physics to finance and biology.

To solve this type of differential equation, one effective method is the separation of variables. This involves rearranging the equation such that all terms involving y are on one side and all terms involving t are on the other. After separation, you integrate both sides. This technique relies on the understanding that the integral of a derivative can lead back to the original function, up to an added constant. Applying this to our equation, integration yields a natural logarithm on one side, which can then be exponentiated to solve explicitly for the function y.

The result of this integration shows the exponential nature of the solution, typically expressed as $y = y_0 e^{kt}$, where $y_0$ represents the initial condition or the value of y at t=0. Understanding this solution represents an essential skill in predicting how systems evolve over time and is fundamental to both theory and practical applications in modeling real-world scenarios.