Which of the following are linear first order differential equations?

(A) $\frac{dy}{dx} + P(x)y = Q(x)$

(B) $\frac{dx}{dy} + P(y)x = Q(y)$

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

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

Choose the correct answer from the options given below:

A linear first-order differential equation must satisfy these conditions:

1. First-order: Only the first derivative appears (like $\frac{dy}{dx}$), no higher derivatives
2. Linear: The dependent variable and its derivative appear only to the power of 1
3. Standard form: Can be written as $\frac{dy}{dx} + P(x)y = Q(x)$

There should be no products like $y \cdot \frac{dy}{dx}$, no powers like $y^2$ or $(\frac{dy}{dx})^2$, and the dependent variable should not appear in denominators.

---

(A) $\frac{dy}{dx} + P(x)y = Q(x)$

This is the standard form itself.

First-order: only $\frac{dy}{dx}$ appears

Linear: $y$ appears to power 1 only

This is a linear first-order differential equation.

---

(B) $\frac{dx}{dy} + P(y)x = Q(y)$

This has $x$ and $y$ roles swapped, where $x$ is the dependent variable and $y$ is the independent variable.

First-order: only $\frac{dx}{dy}$ appears

Linear in $x$: $x$ appears to power 1 only

This is a linear first-order differential equation.

---

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

Expanding:

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

The term $y\frac{dy}{dx}$ is a product of $y$ and $\frac{dy}{dx}$, which violates the linearity condition.

This is not a linear first-order differential equation.

---

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

Dividing everything by $(1 + x^2)$:

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

This is in the form $\frac{dy}{dx} + P(x)y = Q(x)$ where $P(x) = \frac{2x}{1+x^2}$ and $Q(x) = \frac{2}{1+x^2}$

First-order: only $\frac{dy}{dx}$ appears

Linear: $y$ appears to power 1 only

This is a linear first-order differential equation.

---

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