Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Differential Equation | Degree | ||
| (A) | \(xy\displaystyle\frac{d^2y}{dx^2}+x\left(\frac{dy}{dx}\right)^2-y\frac{dy}{dx}=0\) | (I) | 3 |
| (B) | \(\displaystyle\frac{d^2y}{dx^2}+\log\left(\frac{dy}{dx}\right)=0\) | (II) | 1 |
| (C) | \(\left(\displaystyle\frac{d^2y}{dx^2}\right)^2+\left(\displaystyle\frac{dy}{dx}\right)^3+\displaystyle\frac{dy}{dx}+1=0\) | (III) | not defined |
| (D) | \(2x^2\left(\displaystyle\frac{d^2y}{dx^2}\right)^3-5\left(\displaystyle\frac{dy}{dx}\right)^2+y=0\) | (IV) | 2 |
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
The degree of a differential equation is the highest power of the highest-order derivative, and it is defined only when the equation is a polynomial in the derivatives. In all four equations the highest-order derivative is \(\frac{d^2y}{dx^2}\).
(A) \(xy\frac{d^2y}{dx^2}+x\left(\frac{dy}{dx}\right)^2-y\frac{dy}{dx}=0\)
This matches (II) 1, since \(\frac{d^2y}{dx^2}\) appears only to the power 1.
(B) \(\frac{d^2y}{dx^2}+\log\left(\frac{dy}{dx}\right)=0\)
This matches (III) not defined, since \(\log\left(\frac{dy}{dx}\right)\) is not a polynomial in the derivatives.
(C) \(\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^3+\frac{dy}{dx}+1=0\)
This matches (IV) 2, since \(\frac{d^2y}{dx^2}\) appears to the power 2. The power 3 belongs to \(\frac{dy}{dx}\), which is of lower order.
(D) \(2x^2\left(\frac{d^2y}{dx^2}\right)^3-5\left(\frac{dy}{dx}\right)^2+y=0\)
This matches (I) 3, since \(\frac{d^2y}{dx^2}\) appears to the power 3.
So the matching is (A)-(II), (B)-(III), (C)-(IV), (D)-(I).