Which of the following are linear first order differential equations?
(A) \(\displaystyle\frac{dy}{dx}+ P(x)y = Q(x)\)
(B) \(\displaystyle\frac{dx}{dy}+ P(y)x= Q(y)\)
(C) \((x -y)\displaystyle\frac{dy}{dx}= x + 2y\)
(D) \((1+x^2)\displaystyle\frac{dy}{dx}+ 2xy = 2\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
A first order equation is linear if it can be written as \(\displaystyle\frac{dy}{dx} + P(x)y = Q(x)\), or as \(\displaystyle\frac{dx}{dy} + P(y)x = Q(y)\) with x as the dependent variable: the dependent variable and its derivative appear only to the first power and are not multiplied together.
Checking the statements
(A) \(\displaystyle\frac{dy}{dx}+ P(x)y = Q(x)\) — Correct. This is the standard linear form in y.
(B) \(\displaystyle\frac{dx}{dy}+ P(y)x= Q(y)\) — Correct. This is the standard linear form with x as the dependent variable.
(C) \((x -y)\displaystyle\frac{dy}{dx}= x + 2y\) — Incorrect. The term \(-y\displaystyle\frac{dy}{dx}\) is a product of y and its derivative, so the equation is not linear in y. Solved for the derivative,
\( \displaystyle \frac{dy}{dx} = \frac{x + 2y}{x - y} \)
with y in the denominator (it is a homogeneous equation). It is not linear in x either, since \(\displaystyle\frac{dx}{dy} = \frac{x - y}{x + 2y}\).
(D) \((1+x^2)\displaystyle\frac{dy}{dx}+ 2xy = 2\) — Correct. Dividing by \(1 + x^2\):
\( \displaystyle \frac{dy}{dx} + \frac{2x}{1 + x^2}\,y = \frac{2}{1 + x^2} \)