Which of the following differential equations can be written in the variable separable form \(\displaystyle\frac{dy}{dx} = g(x)\,h(y)\)?
(A) \(\displaystyle\frac{dy}{dx} = e^{x - y}\)
(B) \(\displaystyle\frac{dy}{dx} = x^2 + y^2\)
(C) \(\displaystyle\frac{dy}{dx} = \sin (x + y) + \sin (x - y)\)
(D) \(\displaystyle\frac{dy}{dx} = \cos (x + y)\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
An equation is in variable separable form when its right side is a product of a function of x alone and a function of y alone.
\( e^{x - y} = e^{x} \cdot e^{-y} \)
Using \(\sin (x + y) + \sin (x - y) = 2\sin x\cos y\):
\( \sin (x + y) + \sin (x - y) = 2\sin x \cdot \cos y \)
On the other hand,
\( \cos (x + y) = \cos x\cos y - \sin x\sin y \)
is a difference of two products and does not reduce to one product, and \(x^2 + y^2\) is a sum of a function of x and a function of y.
So (A) and (C) are products of a function of x and a function of y, while (B) and (D) are not.
Checking the statements
(A) \(\frac{dy}{dx} = e^{x - y}\) — Correct. It is \(e^{x} \cdot e^{-y}\).
(B) \(\frac{dy}{dx} = x^2 + y^2\) — Incorrect. A sum like \(x^2 + y^2\) cannot be written as one product g(x)h(y).
(C) \(\frac{dy}{dx} = \sin (x + y) + \sin (x - y)\) — Correct. It equals \(2\sin x \cdot \cos y\).
(D) \(\frac{dy}{dx} = \cos (x + y)\) — Incorrect. It is \(\cos x\cos y - \sin x\sin y\), not a single product g(x)h(y).