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

To derive the differential equation for \( y = e^{rx} \), we first need to take the derivative of \( y \) with respect to \( x \). The first derivative, \( \frac{dy}{dx} \), is \( r e^{rx} \). We set up a relationship between \( y \) and its derivative: since \( y = e^{rx} \), we can express this as \( \frac{dy}{dx} = r y \). The resulting first-order linear differential equation is \( \frac{dy}{dx} - ry = 0 \). Now, if you’re looking to solve this equation, you can use the method of separation of variables or an integrating factor! Just remember to keep track of your algebraic manipulations. A common mistake is forgetting to apply the rules of exponentials properly, so double-check your work!