Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(II), (B)-(III), (C)-(IV), (D)-(I)

The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)

Matching the items with explanation:

(A) $\frac{d^2y}{dx^2} = e^{\frac{dy}{dx}}$:

The degree is not defined because the derivative $\frac{dy}{dx}$ is inside the exponential function, which makes the equation non-polynomial. → (II) not defined.

(B) $\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0$:

The order is the highest derivative present, which is $\frac{d^3y}{dx^3}$. The degree is the exponent of this highest-order derivative, which is 1. Thus, the order is $3$. → (III) 3.

(C) $\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 - 5x^2 = 0$:

The highest-order derivative is $\frac{d^2y}{dx^2}$, which appears with power 1. Hence, the degree is 2 due to the squared first derivative. → (I) 2.

(D) $\frac{dy}{dx} + 3y = e^x$:

The order is 1, and the degree is also 1 since the first derivative appears linearly. $p + q = 1 + 1 = 2$. → (IV) 1.