Question

The degree of the differential equation $\log \left( \frac{dy}{dx} \right) = \left( 2x + 3 \frac{dy}{dx} \right)^2$ is

• A. 1
• B. 2
• C. 3
• D. not defined

Hint:

The degree of a differential equation is only defined if the equation is a polynomial in its derivatives. If the derivatives appear as arguments of transcendental functions (like $\sin, \cos, \tan, \log, e$, etc.), then the degree is not defined.

Answer 1

The Correct Option is D

Step 1: Identify the highest-order derivative.
The given differential equation is $\log \left( \frac{dy}{dx} \right) = \left( 2x + 3 \frac{dy}{dx} \right)^2$. The highest-order derivative present in this equation is $\frac{dy}{dx}$, which is the first derivative.
Step 2: Check if the equation is a polynomial in its derivatives.
The degree of a differential equation is defined as the highest power of the highest-order derivative, provided the equation is a polynomial in its derivatives (i.e., it can be expressed as a polynomial in $\frac{dy}{dx}, \frac{d^2y}{dx^2}$, etc., free from radicals and fractions of these derivatives). In this equation, we have the term $\log \left( \frac{dy}{dx} \right)$. The presence of a logarithmic function of a derivative means that the equation is not a polynomial in its derivatives.
Step 3: Determine the degree.
Since the differential equation involves a transcendental function (logarithm) of the derivative $\frac{dy}{dx}$, the degree of the differential equation is not defined. Thus, the degree of the differential equation $\log \left( \frac{dy}{dx} \right) = \left( 2x + 3 \frac{dy}{dx} \right)^2$ is $\boxed{\text{not defined}}$.