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:
Answer & explanation
Correct answer: option 1
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)