Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Statement | Its value | ||
| (A) | Degree of the differential equation \(\displaystyle\frac{d^2y}{dx^2} = e^{dy/dx}\) is | (I) | 2 |
| (B) | Order of the differential equation \(\left(\displaystyle\frac{dy}{dx}\right)^2 + \displaystyle\frac{d^3y}{dx^3} = 0\) is | (II) | not defined |
| (C) | Degree of the differential equation \(\displaystyle\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 - 5x^2 = 0\) | (III) | 3 |
| (D) | If \(p\) is the order and \(q\) is the degree of the differential equation \(\displaystyle\frac{dy}{dx} + 3y = e^x\), then \(p + q\) is | (IV) | 1 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(IV), (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) Degree of \(\frac{d^2y}{dx^2} = e^{dy/dx}\)
This matches (II) not defined. The derivative \(\frac{dy}{dx}\) appears inside an exponential, so the equation is not a polynomial in the derivatives.
(B) Order of \(\left(\frac{dy}{dx}\right)^2 + \frac{d^3y}{dx^3} = 0\)
This matches (III) 3. The highest derivative is \(\frac{d^3y}{dx^3}\).
(C) Degree of \(\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 - 5x^2 = 0\)
This matches (IV) 1. The highest derivative \(\frac{d^2y}{dx^2}\) occurs to the power 1; the square on \(\frac{dy}{dx}\) does not count.
(D) p + q for \(\frac{dy}{dx} + 3y = e^x\)
This matches (I) 2. The highest derivative is \(\frac{dy}{dx}\), to the power 1:
\(p = 1,\quad q = 1,\quad p + q = 2\)
So the matching is (A)-(II), (B)-(III), (C)-(IV), (D)-(I).