Consider the following differential equations:
(A) \(\displaystyle\frac{d^3y}{dx^3} + 2\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\)
(B) \(\left(\displaystyle\frac{dy}{dx}\right)^2 + y = e^x\)
(C) \(\displaystyle\frac{d^2y}{dx^2} + \sin y = 0\)
(D) \(\displaystyle\frac{d^2y}{dx^2} + \cos\left(\frac{dy}{dx}\right) = 0\)
Which of the above differential equations have degree 1?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (C) only
The degree is the power of the highest derivative, and it is defined only when the equation is a polynomial in the derivatives. Functions of y or x alone, such as \(\sin y\) or \(e^x\), do not matter; a function such as \(\cos\left(\frac{dy}{dx}\right)\) of a derivative does.
(A) The highest derivative \(\frac{d^3y}{dx^3}\) has power 1.
(B) The only derivative \(\frac{dy}{dx}\) has power 2.
(C) The highest derivative \(\frac{d^2y}{dx^2}\) has power 1, and \(\sin y\) contains no derivative.
(D) \(\cos\left(\frac{dy}{dx}\right)\) is not a polynomial in \(\frac{dy}{dx}\).
So the degrees are 1, 2, 1 and not defined respectively.
Checking the statements
(A) \(\frac{d^3y}{dx^3} + 2\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0\) — Correct. Degree 1: \(\frac{d^3y}{dx^3}\) has power 1.
(B) \(\left(\frac{dy}{dx}\right)^2 + y = e^x\) — Incorrect. Degree 2: \(\frac{dy}{dx}\) has power 2.
(C) \(\frac{d^2y}{dx^2} + \sin y = 0\) — Correct. Degree 1: \(\sin y\) has no derivative in it.
(D) \(\frac{d^2y}{dx^2} + \cos\left(\frac{dy}{dx}\right) = 0\) — Incorrect. Degree not defined: \(\cos\left(\frac{dy}{dx}\right)\) is not a polynomial in \(\frac{dy}{dx}\).