Question

Find the order and degree (if defined) of the following differential equations:

$\left(\frac{d^2 y}{dx^2}\right)=x \log \left(\frac{dy}{dx}\right)$

• A. (1, 2)
• B. (2, 1)
• C. (2, 2)
• D. None

Answer 1

Answer: (D)

The order of the differential equation is 2, and the degree is not defined.

Answer 2

• Understanding Order and Degree of a Differential Equation:

  • Order: The order of a differential equation is the order of the highest derivative present in the equation.
  • Degree: The degree of a differential equation is the power of the highest order derivative when the differential equation is expressed as a polynomial in derivatives. For the degree to be defined, the differential equation must be free from radicals and fractions involving derivatives, and all derivatives must appear as polynomials (i.e., not inside transcendental functions like log, sin, cos, exponential, etc.).

• Analyzing the given differential equation: The given differential equation is: $\left(\frac{d^2 y}{dx^2}\right)=x \log \left(\frac{dy}{dx}\right)$

• Finding the Order: The derivatives present in the equation are $\frac{d^2 y}{dx^2}$ (second order) and $\frac{dy}{dx}$ (first order). The highest order derivative is $\frac{d^2 y}{dx^2}$. Therefore, the order of the differential equation is 2.

• Finding the Degree: For the degree to be defined, the differential equation must be a polynomial in its derivatives. In the given equation, the term $\log \left(\frac{dy}{dx}\right)$ involves the first derivative $\frac{dy}{dx}$ inside a logarithmic function. Since the derivative $\frac{dy}{dx}$ is present inside a transcendental function (logarithm), the differential equation cannot be expressed as a polynomial in its derivatives. Therefore, the degree of this differential differential equation is not defined.

• Conclusion: The order of the differential equation is 2, and the degree is not defined.