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), (B) and (D) only

The correct answer is Option (1) → (A), (B) and (D) only

Linear first-order differential equations must be expressible in the form

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

Check each option:

(A) $\frac{dy}{dx}+P(x)y=Q(x)$ → already linear. ✔

(B) $\frac{dx}{dy}+P(y)x=Q(y)$ → linear in $x$ (dependent variable is $x$). ✔

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

Rewrite:

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

This becomes

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

The coefficients depend on both $x$ and $y$ in a nonlinear way → NOT linear. ✘

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

Divide by $(1+x^{2})$:

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

This matches the linear form. ✔

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