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LearnAQA7367First teaching 2017Paper 1 · Compulsory contentDifferential equations

Paper 1 · Module

Differential equations

2 subtopics≈ 90 min in total

Start the module: First-order equations: integrating factors Open graphing calculator

First-order equations: integrating factors
An equation $\frac{dy}{dx}+P(x)y=Q(x)$ can't usually be separated. Multiply through by the integrating factor $I=e^{\int P\,dx}$ : the left side becomes $\frac{d}{dx}(Iy)$ , exactly the derivative of a product, so one integration solves it.
≈ 40 min
Second-order equations: auxiliary equation
For $ay''+by'+cy=f(x)$ , try $y=e^{mx}$ in the "homogeneous" equation: $am^2+bm+c=0$ (the auxiliary equation). Its roots give the complementary function. Add a particular integral matching $f(x)$ , then fix the two constants with two conditions.
≈ 50 min