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The differential equation for \( y = e^{r x} \) is \( \frac{dy}{dx} - r y = 0 \).
Solución
The given function is y = e^(r x).
To find its differential equation, differentiate y with respect to x:
dy/dx = r e^(r x).
Since y = e^(r x), we can substitute back to obtain:
dy/dx = r y.
This can be rearranged to form the differential equation:
dy/dx - r y = 0.
Thus, y = e^(r x) is a solution of the differential equation dy/dx - r y = 0.
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The function \( y = e^{r x} \) satisfies the differential equation \( \frac{dy}{dx} = r y \). This is a first-order linear ordinary differential equation where the derivative of \( y \) is proportional to \( y \) itself, with \( r \) being the constant of proportionality. To solve it, you can use the method of separation of variables or apply the characteristic equation. Either way, the exciting part is that it leads you right back to the original exponential function, showcasing the unique and fascinating properties of exponential growth in mathematics!
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