Consider the following statements:
(A) The order of a differential equation is always a positive integer.
(B) The degree of a differential equation, when defined, is always a positive integer.
(C) The degree of the differential equation \(\displaystyle\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^3\) is 3.
(D) The differential equation \(\left(\displaystyle\frac{dy}{dx}\right)^2 + \displaystyle\frac{dy}{dx} - x = 0\) is of order 1 and degree 2.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (D) only
The order is the order of the highest derivative, so it is a positive integer. The degree, when the equation is a polynomial in its derivatives, is the highest power of the highest derivative, so it is also a positive integer.
In \(\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^3\) the highest derivative is \(\frac{d^2y}{dx^2}\), with power 1.
In \(\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} - x = 0\) the only derivative is \(\frac{dy}{dx}\), with highest power 2.
So order and degree (when defined) are positive integers, the first equation has degree 1, and the second has order 1 and degree 2.
Checking the statements
(A) The order of a differential equation is always a positive integer. — Correct. It is the order 1, 2, 3, ... of the highest derivative.
(B) The degree of a differential equation, when defined, is always a positive integer. — Correct. It is a power of the highest derivative in a polynomial equation.
(C) The degree of the differential equation \(\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^3\) is 3. — Incorrect. The highest derivative \(\frac{d^2y}{dx^2}\) has power 1, so the degree is 1.
(D) The differential equation \(\left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} - x = 0\) is of order 1 and degree 2. — Correct. Only \(\frac{dy}{dx}\) appears, with highest power 2.