Question

Find the differential equation of y = $e^{mx}$

• A. $\frac{dy}{dx} - \frac{ylogy}{x} = 0$
• B. $\frac{dy}{dx} = 0$
• C. $\frac{dy}{dx} + \frac{ylogy}{x} = 0$
• D. $\frac{dy}{dx} - \frac{ylogy}{xy} = 0$

Answer 1

Answer: (A)

$\frac{dy}{dx} - \frac{ylogy}{x} = 0$

Answer 2

To find the differential equation for the given function $y = e^{mx}$, we need to eliminate the arbitrary constant 'm'.

1. Given function: $y = e^{mx} \quad \cdots (1)$

2. Differentiate equation (1) with respect to x: $\frac{dy}{dx} = \frac{d}{dx}(e^{mx})$ $\frac{dy}{dx} = m e^{mx} \quad \cdots (2)$

3. Substitute $y$ from equation (1) into equation (2): From (1), we know $e^{mx} = y$. So, $\frac{dy}{dx} = my \quad \cdots (3)$

4. Eliminate 'm' using equation (1): Take the natural logarithm (ln) on both sides of equation (1): $\ln y = \ln (e^{mx})$ $\ln y = mx$ Now, solve for 'm': $m = \frac{\ln y}{x} \quad \cdots (4)$

5. Substitute the expression for 'm' from equation (4) into equation (3): $\frac{dy}{dx} = \left(\frac{\ln y}{x}\right) y$ $\frac{dy}{dx} = \frac{y \ln y}{x}$

6. Rearrange the equation to match the given options: $\frac{dy}{dx} - \frac{y \ln y}{x} = 0$

Assuming $\log y$ in the options refers to $\ln y$ (natural logarithm), this matches the first option.