Match the LIST-I with LIST-II
| LIST-I | LIST-II | ||
| A. | The degree of the differential equation \( \left(\displaystyle\frac{d^{3}y}{dx^{3}}\right)^{4}+\left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{5}+\left(\displaystyle\frac{dy}{dx}\right)+y=0 \) | I. | 0 |
| B. | The order of the differential equation \( \left(\displaystyle\frac{d^{3}y}{dx^{3}}\right)^{4}+\left(\displaystyle\frac{d^{2}y}{dx^{2}}\right)^{5}+\left(\displaystyle\frac{dy}{dx}\right)+y^{2}=0 \) | II. | 4 |
| C. | The number of arbitrary constants in the general solution of a differential equation of second order is | III. | 3 |
| D. | The number of arbitrary constants in a particular solution of a differential equation of third order is | 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
(A) The degree of \( \left(\frac{d^{3}y}{dx^{3}}\right)^{4}+\left(\frac{d^{2}y}{dx^{2}}\right)^{5}+\frac{dy}{dx}+y=0 \)
This matches (II) 4. The equation is a polynomial in its derivatives; the degree is the power of the highest order derivative \( \frac{d^{3}y}{dx^{3}} \), which is 4 (the larger power 5 belongs to a lower order derivative).
(B) The order of \( \left(\frac{d^{3}y}{dx^{3}}\right)^{4}+\left(\frac{d^{2}y}{dx^{2}}\right)^{5}+\frac{dy}{dx}+y^{2}=0 \)
This matches (III) 3. The highest derivative present is \( \frac{d^{3}y}{dx^{3}} \); the \( y^{2} \) term does not affect the order.
(C) Arbitrary constants in the general solution of a second order equation
This matches (IV) 2. The general solution of an equation of order n has n arbitrary constants.
(D) Arbitrary constants in a particular solution of a third order equation
This matches (I) 0. A particular solution is obtained by giving definite values to all the constants, so it has none.
So the matching is (A)-(II), (B)-(III), (C)-(IV), (D)-(I).