Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| (A) | Degree of this differential equation \(\displaystyle\frac{d^4y}{dx^4}+2\log_e\left(\frac{d^3y}{dx^3}\right)=0\) | (I) | 1 |
| (B) | Order of this differential equation \(e^{\left(\frac{dy}{dx}\right)^3}+3y\left(\displaystyle\frac{d^2y}{dx^2}\right)^3=0\) | (II) | 4 |
| (C) | Degree of \(\displaystyle\frac{d^4y}{dx^4}+\left(\frac{dy}{dx}\right)^2= 0\) | (III) | not defined |
| (D) | Order of the differential equation \(2\displaystyle\frac{d^4y}{dx^4}+\left(\frac{d^2y}{dx^2}\right)^5=0\) | (IV) | 2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(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, defined only when the equation is a polynomial in the derivatives.
(A) Degree of \(\displaystyle\frac{d^4y}{dx^4}+2\log_e\left(\frac{d^3y}{dx^3}\right)=0\)
This matches (III) not defined. The derivative \(\frac{d^3y}{dx^3}\) appears inside a logarithm, so the equation is not a polynomial in the derivatives.
(B) Order of \(e^{\left(\frac{dy}{dx}\right)^3}+3y\left(\displaystyle\frac{d^2y}{dx^2}\right)^3=0\)
This matches (IV) 2. The highest derivative present is \(\frac{d^2y}{dx^2}\). (The exponential makes the degree undefined, but the order is still 2.)
(C) Degree of \(\displaystyle\frac{d^4y}{dx^4}+\left(\frac{dy}{dx}\right)^2= 0\)
This matches (I) 1. The equation is a polynomial in the derivatives, and the highest derivative \(\frac{d^4y}{dx^4}\) has power 1.
(D) Order of \(2\displaystyle\frac{d^4y}{dx^4}+\left(\frac{d^2y}{dx^2}\right)^5=0\)
This matches (II) 4. The highest derivative present is \(\frac{d^4y}{dx^4}\); the power 5 belongs to a lower derivative and does not matter.
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).