Match List-I with List-II

List-I	List-II
Differential Equation	Order and degree of differential equation
(A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$	(I) Order = 1, Degree = 2
(B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$	(II) Order = 2, Degree = 1
(C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$	(III) Order = 2, Degree = 2
(D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$	(IV) Order = 2, Degree = Not defined

Choose the correct answer from the options given below:

The order of a differential equation is the highest order derivative present, and the degree is the power of that highest order derivative, defined only when the equation is polynomial in its derivatives.

If any derivative appears inside a transcendental function ($e$, $\sin$, $\log$, etc.), the degree is not defined.

---

(A) $\frac{d^2y}{dx^2} + 2\left(\frac{dy}{dx}\right)^2 = e^{\frac{dy}{dx}} + 1$

The highest derivative is $\frac{d^2y}{dx^2}$, so Order $= 2$.

On the right side, $\frac{dy}{dx}$ appears inside an exponential $e^{\frac{dy}{dx}}$, so the equation is not polynomial in derivatives.

Degree $=$ Not Defined

$(A) \rightarrow (IV)$

---

(B) $\left(\frac{d^2y}{dx^2}\right)^2 + 4\left(\frac{dy}{dx}\right)^3 = e^y - 1$

The highest derivative is $\frac{d^2y}{dx^2}$, so Order $= 2$.

$e^y$ involves the dependent variable $y$, not a derivative, so the equation is polynomial in derivatives. The highest order derivative is raised to power $2$.

Degree $= 2$

$(B) \rightarrow (III)$

---

(C) $3\left(\frac{dy}{dx}\right) + 4y + e^y = \frac{dx}{dy}$

The only derivative present is $\frac{dy}{dx}$, so Order $= 1$.

Since $\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$, let $p = \frac{dy}{dx}$:

$3p + 4y + e^y = \frac{1}{p}$

$3p^2 + (4y + e^y)\,p - 1 = 0$

This is polynomial in $p$ with highest power $2$.

Degree $= 2$

$(C) \rightarrow (I)$

---

(D) $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right) = \left(e^y + \frac{dy}{dx}\right)^2$

The highest derivative is $\frac{d^2y}{dx^2}$, so Order $= 2$.

Expanding the right side:

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

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

The $e^y$ terms involve only the dependent variable, not a derivative. The highest order derivative $\frac{d^2y}{dx^2}$ appears with power $1$.

Degree $= 1$

$(D) \rightarrow (II)$

---

List-I	List-II
(A)	(IV) — Order = 2, Degree = Not defined
(B)	(III) — Order = 2, Degree = 2
(C)	(I) — Order = 1, Degree = 2
(D)	(II) — Order = 2, Degree = 1