The order and degree of the differential equation `(d^2y)/(dx^2)+("dy"/"dx")^(1/3)+x^(1/4)=0` are respectively

Options

A. 2,3
B. 3,3
C. 2,6
D. 2,4

Answer

To determine the order and degree of the differential equation

\[ \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^{\frac{1}{3}} + x^{\frac{1}{4}} = 0, \]

we will follow these steps:

### Step 1: Identify the Order of the Differential Equation
The order of a differential equation is defined as the highest order of derivative present in the equation.

In the given equation, we have:
- The term \(\frac{d^2y}{dx^2}\) which is the second derivative of \(y\).
- The term \(\frac{dy}{dx}\) which is the first derivative of \(y\).

The highest derivative present is \(\frac{d^2y}{dx^2}\), which is of order 2.

### Step 2: Identify the Degree of the Differential Equation
The degree of a differential equation is defined as the power of the highest order derivative after the equation has been made polynomial in derivatives.

In our equation, we have:
- The term \(\frac{d^2y}{dx^2}\) which is to the power of 1.
- The term \(\left(\frac{dy}{dx}\right)^{\frac{1}{3}}\) is not a polynomial term because it involves a fractional exponent.

To find the degree, we need to express the equation in a polynomial form. However, since the term \(\left(\frac{dy}{dx}\right)^{\frac{1}{3}}\) is present, we cannot express it as a polynomial.

To make it polynomial, we can introduce a substitution, but since we are looking for the degree directly from the given equation, we note that the degree is not defined due to the presence of the fractional exponent.

### Conclusion
Thus, we conclude:
- The order of the differential equation is **2**.
- The degree of the differential equation is **not defined** (or can be considered as **fractional** due to the term \(\left(\frac{dy}{dx}\right)^{\frac{1}{3}}\)).

### Final Answer
The order is 2 and the degree is not defined (or can be considered as fractional).

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