The integrating factor (IF) is the cornerstone of solving linear first-order DEs. For dy/dx + P(x)y = Q(x), IF = e^(integral P(x) dx).

Common P(x) and their IFs:

Key Observations:

1. The constant of integration in integral P dx is always taken as zero (any particular antiderivative works for the IF).

2. When P(x) = f'$\frac{x}{f}$(x), then IF = f(x). This is because integral f'/f dx = ln|f|, and e^(ln|f|) = |f| ≈ f (taking positive branch).

3. For dx/dy + P(y)x = Q(y), the IF is e^(integral P(y) dy). Choose whichever form (dy/dx or dx/dy) gives a simpler IF.

4. Sometimes the same equation can be solved as linear in x or linear in y. Choose the one with the simpler IF.

IF for Non-Standard Forms:

If M dx + N dy = 0 is not exact, sometimes an IF mu exists:

• If $\frac{dM/dy - dN/dx}{N}$ is a function of x only, say h(x), then mu = e^(integral h(x) dx)
• If $\frac{dN/dx - dM/dy}{M}$ is a function of y only, say k(y), then mu = e^(integral k(y) dy)

These are rarely needed in JEE but useful for advanced problems.