Which method is most appropriate for solving the first‑order differential equation $\frac{dy}{dx}-3y=6x$?

Step-by-step solution

1. Identify the form of the equation: it is of the type $\frac{dy}{dx}+P(x)y=Q(x)$ with $P(x)=-3$ and $Q(x)=6x$.
2. Because the equation is linear in $y$ and $\frac{dy}{dx}$, the integrating‑factor method is the standard technique.
3. Separation of variables cannot be applied because the right‑hand side is not a product of a function of $x$ and a function of $y$.
4. Hence the correct method is the integrating‑factor method for linear first‑order ODEs.

Concept tested: Linear first‑order ODE

Published 16 September 2026 · How this solution was prepared and checked