What is the differintial equetion of \( y=e^{r x} \)

To find the differential equation of \( y = e^{rx} \), we start by differentiating it. The first derivative is \( y' = re^{rx} \), and the second derivative is \( y'' = r^2 e^{rx} \). From this, we can formulate a differential equation: \( y'' - r^2 y = 0 \). This shows that any function of the form \( y = e^{rx} \) is a solution to this equation, illustrating the exponential relationship in solutions to linear differential equations. Moreover, the beauty of such differential equations extends far beyond theory! They appear in many real-world scenarios, like population growth, radioactive decay, and financial modeling. For instance, in finance, the growth of investments compounded continuously can be modeled using similar exponential functions, making these equations vital for predicting future values based on current trends.