Integrating factor recipe: 4 steps for a linear differential equationसमाकलन गुणक की विधि: रैखिक अवकल समीकरण के 4 चरणIntegrating factor ki recipe: linear differential equation ke 4 steps

NCERT 9.4.3 · Class 12 Maths · Chapter 9 · interactive diagram

Parts and what they mean

• Step 1: standard form \frac{dy}{dx} + Py = Q (चरण 1: मानक रूप \frac{dy}{dx} + Py = Q) — P and Q must be constants or functions of x only. Example 15: x\frac{dy}{dx} + 2y = x^2 divided by x gives P = \frac{2}{x}, Q = x.
• Step 2: Integrating Factor e^{\int P\,dx} (चरण 2: समाकलन गुणक e^{\int P\,dx}) — Multiplying by it turns the L.H.S. into \frac{d}{dx}\left(y \cdot \text{I.F.}\right). Here e^{2\log x} = e^{\log x^2} = x^2.
• Step 3: y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.})\,dx + C (चरण 3: y \cdot (\text{I.F.}) = \int Q \cdot (\text{I.F.})\,dx + C) — Integrate both sides. Example 15: \int x^3\,dx = \frac{x^4}{4}.
• Step 4: the general solution (चरण 4: व्यापक हल) — Divide by x^2: y = \frac{x^2}{4} + Cx^{-2} — the C must also be divided, so it becomes Cx^{-2}, not C.
• Then (फिर) — Read off P (the coefficient of y) and Q.
• Then (फिर) — Multiply the whole equation by the I.F.
• Then (फिर) — Make y the subject.
• Other form: \frac{dx}{dy} + P_1 x = Q_1 (दूसरा रूप: \frac{dx}{dy} + P_1 x = Q_1) — When P_1, Q_1 are functions of y only, the roles of x and y swap: \text{I.F.} = e^{\int P_1\,dy} (Examples 16 and 22).

Original diagram drawn by Simple Learning Hub, based on NCERT 9.4.3. Report a mistake