Question

The differential equation corresponding to \(y = e^{mx}\) by eliminating \(m\) is

• A. \(x \frac{dy}{dx} = y \log y\)
• B. \(x \frac{dy}{dx} = \log y\)
• C. \(x \frac{dy}{dx} = (y-1) \log y\)
• D. \(x \frac{dy}{dx} = (y+1) \log y\)

Hint:

When differentiating exponential equations, isolating the constant in terms of logarithms before or after differentiation is a highly efficient way to eliminate variables.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To form a differential equation from a given algebraic relation containing an arbitrary constant, we differentiate the relation and then eliminate the constant using substitution.

Step 2: Detailed Explanation:
We are given the relation:
\[ y = e^{mx} \]
Taking the natural logarithm of both sides:
\[ \log y = \log(e^{mx}) \]
\[ \log y = mx \]
From this, we express the arbitrary constant \(m\) as:
\[ m = \frac{\log y}{x} \quad \text{--- (Equation 1)} \]
Now, we differentiate the original relation \(y = e^{mx}\) with respect to \(x\):
\[ \frac{dy}{dx} = e^{mx} \cdot \frac{d}{dx}(mx) \]
\[ \frac{dy}{dx} = m \cdot e^{mx} \]
Since \(y = e^{mx}\), we can substitute \(y\) back into the derivative:
\[ \frac{dy}{dx} = m y \quad \text{--- (Equation 2)} \]
To eliminate \(m\), substitute the value of \(m\) from Equation 1 into Equation 2:
\[ \frac{dy}{dx} = \left(\frac{\log y}{x}\right) y \]
Multiply both sides by \(x\):
\[ x \frac{dy}{dx} = y \log y \]
This matches Option A.

Step 3: Final Answer:
The differential equation is \(x \frac{dy}{dx} = y \log y\).