Match List – I with List – II.

Choose the correct answer from the options given below:

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

The correct answer is Option (1) → A-III, B-IV, C-II, D-I

A. $\frac{d^2 y}{d x^2}=\left(\frac{d y}{d x}\right)^{\frac{3}{2}}$

To find the degree, we must square both sides to remove the fractional exponent:

$\left(\frac{d^2 y}{d x^2}\right)^2 = \left(\frac{d y}{d x}\right)^3$

• Order: 2
• Degree: 2
• Sum: $2 + 2 = 4$ (Matches III)

B. $2\left(\frac{d^3 y}{d x^3}\right)^2+3\left(\frac{d^2 y}{d x^2}\right)+y\left(\frac{d y}{d x}\right)^2=e^x$

The highest derivative is the third derivative.

• Order: 3
• Degree: 2
• Sum: $3 + 2 = 5$ (Matches IV)

C. $\frac{d y}{d x}+\frac{1}{d y / d x}=3$

Multiply the entire equation by $\frac{dy}{dx}$ to clear the fraction:

$\left(\frac{d y}{d x}\right)^2 + 1 = 3\left(\frac{d y}{d x}\right)$

• Order: 1
• Degree: 2
• Sum: $1 + 2 = 3$ (Matches II)

D. $\frac{d y}{d x}+x^2=5$

This is a standard first-order equation.

• Order: 1
• Degree: 1
• Sum: $1 + 1 = 2$ (Matches I)