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:
| P(x) | IF = e^(integral P dx) |
|---|---|
| k (constant) | e^(kx) |
| k/x | x^k (or |x|^k) |
| -k/x | x^(-k) |
| tan x | sec x |
| -tan x | cos x |
| cot x | sin x |
| -cot x | csc x |
| 2x | e^(x^2) |
| n*x^(n-1)/(1+x^n) | 1+x^n |
| 1/(x ln x) | ln x |
| sec^2 x | e^(tan x) |
| 1/(1+x^2) | e^(arctan x) |
Key Observations:
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The constant of integration in integral P dx is always taken as zero (any particular antiderivative works for the IF).
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When P(x) = f'(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).
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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.
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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 (dM/dy - dN/dx)/N is a function of x only, say h(x), then mu = e^(integral h(x) dx)
- If (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.