Integrating Factor Method
The integrating factor method is a standard technique for solving first-order linear ordinary differential equations of the form $$ y' + A(x)\, y = B(x) $$ It involves multiplying both sides of the equation by a specially chosen function known as the integrating factor: $$ M(x) = e^{\int A(x)\, dx} $$
The key idea is to turn the left-hand side of the equation into the derivative of a product, by applying the product or quotient rule in reverse:
$$ f'g + fg' = (f \cdot g)' $$
$$ \frac{f'g - fg'}{g^2} = \left( \frac{f}{g} \right)' $$
Once written as a derivative, the equation can be integrated directly with respect to \( x \) to find the solution.
A Worked Example
Consider the differential equation:
$$ y' - \frac{2y}{x} = 0 $$
This is a first-order homogeneous linear equation, which fits the standard form \( y' + A(x)\, y = B(x) \), with \( A(x) = -\frac{2}{x} \) and \( B(x) = 0 \).
We begin by computing the integrating factor:
$$ M(x) = e^{\int A(x)\, dx} $$
$$ M(x) = e^{\int -\frac{2}{x}\, dx} $$
$$ M(x) = e^{-2 \int \frac{1}{x}\, dx} $$
Since \( \int \frac{1}{x}\, dx = \log x \), we obtain:
$$ M(x) = e^{-2 \log x} = x^{-2} $$
Equivalently:
$$ M(x) = \frac{1}{x^2} $$
Now we multiply both sides of the original equation by the integrating factor:
$$ \left[ y' - \frac{2y}{x} \right] \cdot \frac{1}{x^2} = 0 $$
Expanding the left-hand side:
$$ \frac{y'}{x^2} - \frac{2y}{x^3} = 0 $$
Bringing to common denominator:
$$ \frac{y' x^3 - 2y x^2}{x^5} = 0 $$
Factoring out \( x \):
$$ \frac{x (y' x^2 - 2y x)}{x^5} = 0 $$
Reducing:
$$ \frac{y' x^2 - 2y x}{x^4} = 0 $$
This expression is the result of applying the quotient rule in reverse, where \( f = y \) and \( g = x^2 \):
$$ \left( \frac{y}{x^2} \right)' = 0 $$
We now integrate both sides with respect to \( x \):
$$ \int \left( \frac{y}{x^2} \right)' dx = \int 0 \, dx $$
Which yields:
$$ \frac{y}{x^2} = c $$
Finally, solving for \( y \):
$$ y = c \cdot x^2 $$
This is the general solution of the original differential equation.
And so on.
