Linear First-Order ODEs and the Integrating Factor

A linear first-order ODE has the form

y' + P(x)y = Q(x)

— the unknown function

y

appears linearly (no

y^2

, no

\sin y

). This class includes many financially important equations that separable equations cannot handle: a portfolio that earns interest at rate

r

but pays out a fixed coupon

C

per year satisfies

P' = rP - C

(not separable); the Ornstein-Uhlenbeck mean-reversion ODE

dx/dt = a(b-x)

rewrites as the linear form

x' + ax = ab

. The solution method is the integrating factor

\mu(x) = e^{\int P(x)\,dx}

— a function that collapses the left side into the derivative of a product and turns the equation into a straightforward integration.

Learning objectives. By the end of this note you should be able to:

recognise a first-order linear ODE and put it in standard form $y' + P(x)y = Q(x)$ ;
compute the integrating factor $\mu(x) = e^{\int P\,dx}$ ;
multiply through, recognise the product-rule pattern, and integrate to find the general solution;
apply an initial condition to pin the constant;
solve the portfolio-with-withdrawals and OU mean-reversion ODEs.

In this lesson

The full lesson, with worked examples and exercise solutions, is for members.