Match List-I with List-II
| List-I | List-II | ||
| Differential equation | Order and Degree | ||
| (A) | \( \left(\displaystyle\frac{dy}{dx}\right)^2+3y\left(\displaystyle\frac{d^2y}{dx^2}\right)=0 \) | (I) | 1,1 |
| (B) | \( \sqrt{1-\left(\displaystyle\frac{dy}{dx}\right)^2}=a\left(\displaystyle\frac{d^2y}{dx^2}\right)^{1/3} \) | (II) | 1,2 |
| (C) | \( y=\displaystyle\frac{dy}{dx}+\frac{5}{\left(dy/dx\right)} \) | (III) | 2,1 |
| (D) | \( \sqrt{1-y^2}\,dx+\sqrt{1-x^2}\,dy=0 \) | (IV) | 2,2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) - (III), (B) - (IV), (C) - (II), (D) - (I)
The order is the order of the highest derivative. The degree is the power of that highest derivative once the equation is a polynomial in the derivatives (no radicals, fractional powers or derivatives in a denominator).
(A) \( \left(\frac{dy}{dx}\right)^2+3y\left(\frac{d^2y}{dx^2}\right)=0 \)
This matches (III) 2,1.
The highest derivative is \( \frac{d^2y}{dx^2} \), and it appears to the power 1. The square is on \( \frac{dy}{dx} \), which is not the highest derivative.
(B) \( \sqrt{1-\left(\frac{dy}{dx}\right)^2}=a\left(\frac{d^2y}{dx^2}\right)^{1/3} \)
This matches (IV) 2,2.
Squaring both sides and then cubing both sides:
\( \displaystyle 1-\left(\frac{dy}{dx}\right)^2=a^2\left(\frac{d^2y}{dx^2}\right)^{2/3} \)
\( \displaystyle \left[1-\left(\frac{dy}{dx}\right)^2\right]^3=a^6\left(\frac{d^2y}{dx^2}\right)^2 \)
The highest derivative \( \frac{d^2y}{dx^2} \) now has power 2.
(C) \( y=\frac{dy}{dx}+\frac{5}{(dy/dx)} \)
This matches (II) 1,2.
Multiplying by \( \frac{dy}{dx} \):
\( \displaystyle y\,\frac{dy}{dx}=\left(\frac{dy}{dx}\right)^2+5 \)
Only the first derivative occurs, and its highest power is 2.
(D) \( \sqrt{1-y^2}\,dx+\sqrt{1-x^2}\,dy=0 \)
This matches (I) 1,1.
\( \displaystyle \frac{dy}{dx}=-\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}} \)
The radicals contain only x and y, not the derivative, so the order and the degree are both 1.
So the matching is (A)-(III), (B)-(IV), (C)-(II), (D)-(I).
Tip: Clear radicals and fractional powers only when they contain a derivative, as in (B). Roots of x or y alone, as in (D), do not affect the degree.