11. Which of the following are linear first order differential equations?

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

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

(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:

The correct option is (A), (B) and (D) only.

To determine which of the given differential equations are linear first-order differential equations, let us first understand the general forms of such equations:

1. A first-order differential equation is linear in the dependent variable $y$ (with independent variable $x$) if it can be written in the form:
$$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$$
where $P\left(x\right)$ and $Q\left(x\right)$ are functions of $x$ only.

2. A first-order differential equation is linear in the dependent variable $x$ (with independent variable $y$) if it can be written in the form:
$$\frac{dx}{dy}+P\left(y\right)x=Q\left(y\right)$$
where $P\left(y\right)$ and $Q\left(y\right)$ are functions of $y$ only.

Now, let us analyze each given equation step-by-step:

Analysis of (A):
The equation is:
$$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$$
This matches the standard definition of a linear first-order differential equation in $y$. Therefore, (A) is linear.

Analysis of (B):
The equation is:
$$\frac{dx}{dy}+P\left(y\right)x=Q\left(y\right)$$
This matches the standard definition of a linear first-order differential equation in $x$. Therefore, (B) is linear.

Analysis of (C):
The equation is:
$$(x-y)\frac{dy}{dx}=x+2y$$
If we divide by $(x-y)$, we obtain:
$$\frac{dy}{dx}=\frac{x+2y}{x-y}$$
Since $y$ appears in the denominator, this equation cannot be rearranged into the form $\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$.
If we check linearity in $x$ by taking the reciprocal:
$$\frac{dx}{dy}=\frac{x-y}{x+2y}$$
Here, $x$ appears in both the numerator and the denominator, meaning it cannot be written in the form $\frac{dx}{dy}+P\left(y\right)x=Q\left(y\right)$. Thus, (C) is non-linear.

Analysis of (D):
The equation is:
$$(1+x^2)\frac{dy}{dx}+2xy=2$$
To see if it fits the linear format, we divide the entire equation by the coefficient of the derivative, $(1+x^2)$:
$$\frac{dy}{dx}+\left[\frac{2x}{1+x^2}\right]y=\frac{2}{1+x^2}$$
This perfectly matches the standard linear first-order differential equation form:
$$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$$
where $P\left(x\right)=\frac{2x}{1+x^2}$ and $Q\left(x\right)=\frac{2}{1+x^2}$ are both functions of $x$ only. Therefore, (D) is linear.

Conclusion:
Equations (A), (B), and (D) are linear first-order differential equations, whereas (C) is not.