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 first-order linear differential equation has the following characteristics:

• Only first derivatives appear
• The dependent variable and its derivative appear to the power of 1 only
• Can be written in the form $\frac{dy}{dx} + P(x) \cdot y = Q(x)$ or $\frac{dx}{dy} + P(y) \cdot x = Q(y)$
• $P$ and $Q$ must be functions of the independent variable only

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Option (A): $\frac{dx}{dy} + P_1(y)x = Q_1(y)$

This is already in standard form where $x$ is the dependent variable and $y$ is independent.

The equation is first order and linear in $x$.

Option (A) is a first-order linear differential equation.

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Option (B): $\frac{dy}{dx} + P_2(x)y = Q_2(x)$

This is already in standard form where $y$ is the dependent variable and $x$ is independent.

The equation is first order and linear in $y$.

Option (B) is a first-order linear differential equation.

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Option (C): $(x + y)\frac{dy}{dx} = x - 2y$

Dividing both sides by $(x + y)$:

$\frac{dy}{dx} = \frac{x - 2y}{x + y}$

$\frac{dy}{dx} = \frac{x}{x+y} - \frac{2y}{x+y}$

$\frac{dy}{dx} + \frac{2y}{x+y} = \frac{x}{x+y}$

The coefficient of $y$ is $\frac{2}{x+y}$, which contains both $x$ and $y$.

For a linear equation, the coefficient must be a function of $x$ only.

Option (C) is not a first-order linear differential equation.

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

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

Dividing both sides by $(1 + x^2)$:

$\frac{dy}{dx} = \frac{2xy + x^2 + 3}{1 + x^2}$

$\frac{dy}{dx} = \frac{2xy}{1 + x^2} + \frac{x^2 + 3}{1 + x^2}$

$\frac{dy}{dx} - \frac{2x}{1 + x^2}y = \frac{x^2 + 3}{1 + x^2}$

Here $P(x) = -\frac{2x}{1 + x^2}$ and $Q(x) = \frac{x^2 + 3}{1 + x^2}$ are both functions of $x$ only.

Option (D) is a first-order linear differential equation.

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Options (A), (B), and (D) are first-order linear differential equations.

Option (C) is not a first-order linear differential equation.

The correct answer is: (A), (B) and (D) only