Which of the following are first order linear differential equations?

(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
(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
(C) $(x + y)\frac{dy}{dx}=x-2y$
(D) $(1+x^2)\frac{dy}{dx}-2xy = x^2+3$

Choose the correct answer from the options given below:

(A), (B) and (D) only

The correct answer is Option (3) → (A), (B) and (D) only

To identify **first order linear differential equations**, recall the standard form of a first-order linear DE:

$\displaystyle \frac{dy}{dx} + P(x)\,y = Q(x)$ or $\displaystyle \frac{dx}{dy} + P(y)\,x = Q(y)$

Let's analyze each option:

(A) $\displaystyle \frac{dx}{dy} + P_1(y)\,x = Q_1(y)$ → This is a first-order linear DE in $x$ as a function of $y$ .

(B) $\displaystyle \frac{dy}{dx} + P_2(x)\,y = Q_2(x)$ → This is the standard form of a first-order linear DE.

(C)$(x + y)\,\frac{dy}{dx} = x - 2y$ → Rearranged: $\displaystyle \frac{dy}{dx} = \frac{x - 2y}{x + y}$ → Nonlinear due to division involving $y$ in denominator.

(D) $(1 + x^2)\,\frac{dy}{dx} - 2xy = x^2 + 3$ → Rearranged: $\displaystyle \frac{dy}{dx} - \frac{2x}{1 + x^2} y = \frac{x^2 + 3}{1 + x^2}$ → This is in the linear form.

Correct options: (A), (B), (D)