Which of the following are first order linear differential equations?
(A) \(\displaystyle\frac{dx}{dy} + P_1(y)x = Q_1(y)\) : \(P_1(y)\) and \(Q_1(y)\) are functions of \(y\) or constant functions
(B) \(\displaystyle\frac{dy}{dx} + P_2(x)y = Q_2(x)\) : \(P_2(x)\) and \(Q_2(x)\) are functions of \(x\) or constant functions
(C) \((x + y)\displaystyle\frac{dy}{dx} = x - 2y\)
(D) \((1 + x^2)\displaystyle\frac{dy}{dx} - 2xy = x^2 + 3\)
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
A first order linear differential equation has one of the two standard forms
\( \displaystyle \frac{dy}{dx} + P(x)\,y = Q(x) \)
\( \displaystyle \frac{dx}{dy} + P_1(y)\,x = Q_1(y) \)
Rewriting (C):
\( \displaystyle \frac{dy}{dx} = \frac{x - 2y}{x + y} \)
Dividing (D) by \(1 + x^2\):
\( \displaystyle \frac{dy}{dx} - \frac{2x}{1 + x^2}\,y = \frac{x^2 + 3}{1 + x^2} \)
So (A), (B) and (D) are in a standard linear form, and (C) is not.
Checking the statements
(A) \(\frac{dx}{dy} + P_1(y)x = Q_1(y)\) : \(P_1(y)\) and \(Q_1(y)\) are functions of \(y\) or constant functions — Correct. Linear in x, with y as the independent variable.
(B) \(\frac{dy}{dx} + P_2(x)y = Q_2(x)\) : \(P_2(x)\) and \(Q_2(x)\) are functions of \(x\) or constant functions — Correct. This is the standard linear form in y.
(C) \((x + y)\frac{dy}{dx} = x - 2y\) — Incorrect. In \(\frac{dy}{dx} = \frac{x - 2y}{x + y}\), y appears in the denominator, so it cannot be written as \(\frac{dy}{dx} + P(x)y = Q(x)\).
(D) \((1 + x^2)\frac{dy}{dx} - 2xy = x^2 + 3\) — Correct. \(P(x) = -\frac{2x}{1 + x^2}\) and \(Q(x) = \frac{x^2 + 3}{1 + x^2}\).