Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Differential Equation | Order and degree of differential equation | ||
| (A) | \(\displaystyle\frac{d^2y}{dx^2}+2\left(\frac{dy}{dx}\right)^2=e^{\frac{dy}{dx}}+1\) | (I) | Order = 1, Degree = 2 |
| (B) | \(\displaystyle\left(\frac{d^2y}{dx^2}\right)^2+4\left(\frac{dy}{dx}\right)^3=e^y-1\) | (II) | Order = 2, Degree = 1 |
| (C) | \(\displaystyle 3\left(\frac{dy}{dx}\right)+4y+e^y=\frac{dx}{dy}\) | (III) | Order = 2, Degree = 2 |
| (D) | \(\displaystyle\frac{d^2y}{dx^2}+3\left(\frac{dy}{dx}\right)=\left(e^y+\frac{dy}{dx}\right)^2\) | (IV) | Order = 2, Degree = Not defined |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
The order is the order of the highest derivative. The degree is the power of that derivative, and it is defined only when the equation is a polynomial in the derivatives.
(A) \(\frac{d^2y}{dx^2}+2\left(\frac{dy}{dx}\right)^2=e^{\frac{dy}{dx}}+1\)
This matches (IV) Order = 2, Degree = Not defined. The highest derivative is \(\frac{d^2y}{dx^2}\), so the order is 2. The term \(e^{\frac{dy}{dx}}\) is not a polynomial in \(\frac{dy}{dx}\), so the degree is not defined.
(B) \(\left(\frac{d^2y}{dx^2}\right)^2+4\left(\frac{dy}{dx}\right)^3=e^y-1\)
This matches (III) Order = 2, Degree = 2. The equation is a polynomial in the derivatives, and the highest derivative \(\frac{d^2y}{dx^2}\) occurs with power 2.
(C) \(3\left(\frac{dy}{dx}\right)+4y+e^y=\frac{dx}{dy}\)
This matches (I) Order = 1, Degree = 2. Let \(p = \frac{dy}{dx}\), so \(\frac{dx}{dy} = \frac{1}{p}\):
\( \displaystyle 3p + 4y + e^y = \frac{1}{p} \)
Multiplying by p:
\(3p^2 + (4y + e^y)\,p = 1\)
The only derivative is \(\frac{dy}{dx}\), with highest power 2.
(D) \(\frac{d^2y}{dx^2}+3\left(\frac{dy}{dx}\right)=\left(e^y+\frac{dy}{dx}\right)^2\)
This matches (II) Order = 2, Degree = 1. The right side expands to a polynomial in \(\frac{dy}{dx}\):
\( \displaystyle \left(e^y+\frac{dy}{dx}\right)^2 = e^{2y} + 2e^y\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^2 \)
The highest derivative \(\frac{d^2y}{dx^2}\) occurs with power 1.
So the matching is (A)-(IV), (B)-(III), (C)-(I), (D)-(II).
Tip: A function of y such as \(e^y\) does not affect the degree. The degree is undefined only when a derivative sits inside such a function, as in \(e^{\frac{dy}{dx}}\).