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

To find the differential equation for \( y = e^{rx} \), we need to take its derivatives. The first derivative of \( y \) is \( y' = r e^{rx} \). The second derivative is \( y'' = r^2 e^{rx} \). By eliminating the constant \( r \), we can formulate a differential equation: \( y'' - r^2 y = 0 \). However, since \( r \) can be any constant, we can simply express this as \( y'' - \frac{y'}{y} = 0 \) after substitution, making it a second-order linear equation. Did you know that differential equations have been shaping our understanding of the natural world since the 17th century? Mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz laid the groundwork for calculus, which is fundamental for formulating differential equations. This interplay has allowed scientists to model everything from the motion of planets to population dynamics, proving that these equations are not just abstract concepts, but fundamental tools in science! For anyone interested in diving deeper, a classic yet accessible recommendation is “Ordinary Differential Equations” by Morris Tenenbaum and Harry Pollard. This book is designed for beginners and is filled with practical examples that bridge theory and application. It provides a solid foundation, making it easier to tackle real-world problems using differential equations!