Match List I with List II
| LIST I | LIST II | ||
| A. | Differential equation \(\displaystyle\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^{\frac{1}{4}}+x^{\frac{1}{5}}=0\) | I. | order 2, degree 1 |
| B. | \(\displaystyle\frac{dy}{dx}=\frac{x^{\frac{1}{2}}}{y^{\frac{1}{2}}(1+x)^{\frac{1}{2}}}\) | II. | order 2, degree not defined |
| C. | \(\displaystyle\frac{d^2y}{dx^2}=\cos 3x+\sin 3x\) | III. | order 2, degree 4 |
| D. | \(\displaystyle\frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=\log\left(\frac{dy}{dx}\right)\) | IV. | order 1, degree 2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-III, B-IV, C-I, D-II
The order is the order of the highest derivative present. The degree is the power of that highest derivative once the equation is a polynomial in the derivatives; it is not defined when a derivative sits inside a function such as log or sin.
(A) \(\displaystyle\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^{1/4}+x^{1/5}=0\)
This matches (III) order 2, degree 4.
The highest derivative is \(\frac{d^2y}{dx^2}\), so the order is 2. To remove the fractional power, isolate it and raise both sides to the power 4:
\( \displaystyle \frac{d^2y}{dx^2}+x^{1/5} = -\left(\frac{dy}{dx}\right)^{1/4} \)
\( \displaystyle \left(\frac{d^2y}{dx^2}+x^{1/5}\right)^4 = \frac{dy}{dx} \)
Now the equation is a polynomial in the derivatives, and \(\frac{d^2y}{dx^2}\) appears to the power 4. Degree 4.
(B) \(\displaystyle\frac{dy}{dx}=\frac{x^{1/2}}{y^{1/2}(1+x)^{1/2}}\)
This matches (IV) order 1, degree 2, by elimination.
Only \(\frac{dy}{dx}\) appears, so the order is 1, as in (IV). It appears to the power 1 (the square roots are on x and y only, not on the derivative), so the degree is 1, not 2: the printed entry (IV) does not fully fit.
(C) \(\displaystyle\frac{d^2y}{dx^2}=\cos 3x+\sin 3x\)
This matches (I) order 2, degree 1.
The highest derivative \(\frac{d^2y}{dx^2}\) appears to the power 1. Order 2, degree 1.
(D) \(\displaystyle\frac{d^2y}{dx^2}+2\frac{dy}{dx}+y=\log\left(\frac{dy}{dx}\right)\)
This matches (II) order 2, degree not defined.
The order is 2, but \(\frac{dy}{dx}\) appears inside a logarithm, so the equation is not a polynomial in the derivatives and the degree is not defined.
Note: It seems that there is a mistake in the original NTA paper, which prints entry (IV) as “order 1, degree 2” when it should read “order 1, degree 1” to fit equation B. The question can still be solved, because A, C and D each have only one possible match (III, I and II), and Option (2) is the only option containing A-III, C-I and D-II.