Differential equations (Further Pure 2)
Integrating factor
- For a first-order linear equation $\dfrac{dy}{dx} + P(x)\,y = Q(x)$, multiply by the integrating factor:
$$\mu = e^{\int P\,dx}$$
- The left side becomes $\dfrac{d}{dx}(\mu y)$, so you can integrate directly.
- e.g. for $\dfrac{dy}{dx} + 2y = e^x$, $\mu = e^{2x}$.
Practice
For dy/dx + 2y = eˣ, the integrating factor μ = e^(∫2 dx) is:
∫2 dx = 2x, so μ = e^(2x).
Practice
Multiplying by the integrating factor makes the left side d/dx(μy), so you can integrate directly.
That is the whole point of the integrating factor — it makes the left side an exact derivative.
Constant coefficients
- For a linear equation with constant coefficients, the general solution = the complementary function (solve with right side 0, via the auxiliary equation) + a particular integral (any one solution of the full equation).
Practice
For a constant-coefficient linear equation, the general solution is:
General solution = complementary function (RHS = 0) + a particular integral.
You've got it
Key idea
- linear first-order: multiply by integrating factor $\mu = e^{\int P\,dx}$, then integrate
- constant-coefficient: general = complementary function + particular integral
- the CF comes from the auxiliary equation