Match the differential equation (List-I) with its order and degree (List-II)
| List-I | List-II | ||
| (A) | \( \displaystyle\frac{d^{3}y}{dx^{3}}+x^{2}\left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)=0 \) | (I) | order = 3; degree is not defined |
| (B) | \( \displaystyle\frac{d^{3}y}{dx^{3}}+y^{2}+e^{\frac{dy}{dx}}=0 \) | (II) | order = 3; degree = 1 |
| (C) | \( 1-\left(\displaystyle\frac{dy}{dx}\right)^{2}=\left(a\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{\frac{1}{3}} \) | (III) | order = 2; degree = 1 |
| (D) | \( y=x\left(\displaystyle\frac{dy}{dx}\right)^{3}+\left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{4} \) | (IV) | order = 2; degree = 4 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (II), (B) - (I), (C) - (III), (D) - (IV)
The order is the order of the highest derivative. The degree is defined only when the equation is a polynomial in its derivatives, and it is then the power of the highest order derivative.
(A) \( \frac{d^{3}y}{dx^{3}}+x^{2}\left(\frac{d^{2}y}{dx^{2}}\right)=0 \)
This matches (II) order = 3; degree = 1.
The highest derivative \( \frac{d^{3}y}{dx^{3}} \) appears to the power 1; the \( x^{2} \) is a coefficient, not a power of a derivative.
(B) \( \frac{d^{3}y}{dx^{3}}+y^{2}+e^{\frac{dy}{dx}}=0 \)
This matches (I) order = 3; degree is not defined.
The term \( e^{\frac{dy}{dx}} \) is not a polynomial in \( \frac{dy}{dx} \), so the degree is not defined. The order is 3.
(C) \( 1-\left(\frac{dy}{dx}\right)^{2}=\left(a\frac{d^{2}y}{dx^{2}}\right)^{\frac{1}{3}} \)
This matches (III) order = 2; degree = 1.
Cubing both sides removes the fractional power:
\( \displaystyle \left(1-\left(\frac{dy}{dx}\right)^{2}\right)^{3}=a\frac{d^{2}y}{dx^{2}} \)
Now the equation is a polynomial in the derivatives, and \( \frac{d^{2}y}{dx^{2}} \) appears to the power 1.
(D) \( y=x\left(\frac{dy}{dx}\right)^{3}+\left(\frac{d^{2}y}{dx^{2}}\right)^{4} \)
This matches (IV) order = 2; degree = 4.
The highest derivative \( \frac{d^{2}y}{dx^{2}} \) appears to the power 4.
So the matching is (A)-(II), (B)-(I), (C)-(III), (D)-(IV).
Tip: Clear fractional powers of derivatives (as in (C)) before reading the degree; reading the power \( \frac{1}{3} \) as the degree is a common trap. Only a derivative inside ex, sin, log and similar functions makes the degree undefined.