Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | The degree of differential equation \(\displaystyle\frac{d^3y}{dx^3}= e^{\frac{dy}{dx}}\) | (I) | 2 |
| (B) | The order of differential equation \(\left(\displaystyle\frac{dy}{dx}\right)^2 + \displaystyle\frac{d^3y}{dx^3}=0\) | (II) | 4 |
| (C) | The sum of order and degree of differential equation \(\displaystyle\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)+\left(\frac{dy}{dx}\right)^5= x\) | (III) | not defined |
| (D) | The number of arbitrary constants in the general solution of a differential equation of order 2 | (IV) | 3 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
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) The degree of differential equation \(\displaystyle\frac{d^3y}{dx^3}= e^{\frac{dy}{dx}}\)
This matches (III) not defined. The derivative \(\frac{dy}{dx}\) appears inside an exponential, so the equation is not a polynomial in the derivatives.
(B) The order of differential equation \(\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3}=0\)
This matches (IV) 3. The highest derivative is \(\frac{d^3y}{dx^3}\).
(C) The sum of order and degree of differential equation \(\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)+\left(\frac{dy}{dx}\right)^5= x\)
This matches (II) 4. Since
\( \displaystyle \frac{d}{dx}\left(\frac{d^2y}{dx^2}\right) = \frac{d^3y}{dx^3} \)
the equation is \(\frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^5 = x\). Its order is 3, and the highest derivative appears to the power 1, so its degree is 1. The sum is 3 + 1 = 4.
(D) The number of arbitrary constants in the general solution of a differential equation of order 2
This matches (I) 2. The general solution of an equation of order n has exactly n arbitrary constants.
So the matching is (A)-(III), (B)-(IV), (C)-(II), (D)-(I).
Tip: Rewrite the equation before reading its order: \(\frac{d}{dx}\left(\frac{d^2y}{dx^2}\right)\) is a third derivative, not a second.