Introduction
Ordinary differential equations (ODEs) are equations involving an unknown function and its derivatives, and are the natural mathematical language for describing systems that change continuously — from the equations of motion in Dynamics (Chapter III) to circuits, population growth, and cooling. This chapter equips the essential solution techniques.
1. First-order equations: separable, exact, and linear types
A separable equation can be rearranged so that all terms involving the dependent variable (with its differential) are on one side and all terms involving the independent variable are on the other, allowing direct integration of both sides. A linear first-order equation, dy/dx + P(x)y = Q(x), is solved using an integrating factor, μ(x) = e^(∫P dx), which is constructed specifically so that multiplying through by it turns the left-hand side into the derivative of a single product, d(μy)/dx, making the equation directly integrable. An exact equation, M(x,y)dx + N(x,y)dy = 0, satisfies ∂M/∂y = ∂N/∂x, which guarantees the left side is the total differential of some function F(x,y); the solution is then simply F(x,y) = constant. Recognising which of these forms (or none, requiring a substitution to transform it into one) applies to a given equation is the essential first step in solving any first-order ODE.
2. Higher-order linear equations with constant coefficients
For an equation like ay″ + by′ + cy = 0, substituting a trial solution y = e^(mx) converts the differential equation into an algebraic characteristic (auxiliary) equation, am² + bm + c = 0. The nature of its roots determines the solution’s form: two distinct real roots give a sum of two exponential terms; a repeated real root requires multiplying the second solution by x (to keep the two solutions independent); complex conjugate roots (a ± bi) give an oscillating solution combining exponential growth/decay with sine and cosine terms — directly connecting to damped and forced oscillations in Physics Chapter III.
3. Complementary and particular solutions
The general solution of a non-homogeneous linear equation is the sum of the complementary function (the general solution of the corresponding homogeneous equation, found via the characteristic equation above) and a particular integral (any single solution that satisfies the full non-homogeneous equation, typically found using the method of undetermined coefficients — guessing a trial form matching the right-hand side’s functional type, or variation of parameters for more general cases). Both pieces are necessary: the complementary function alone would not satisfy the full equation, and the particular integral alone would not contain the arbitrary constants needed to match arbitrary initial or boundary conditions.
4. Initial and boundary conditions
A general solution to an nth-order ODE contains n arbitrary constants; these are fixed by applying exactly n given conditions. Initial conditions specify the solution (and its derivatives) at a single point (typical for time-dependent physical problems, where conditions are given “at t=0”); boundary conditions specify the solution at two or more different points (typical for spatial problems, connecting directly to the boundary-value problems of partial differential equations in Chapter V). Conditions must always be applied to the general solution (after combining complementary and particular parts), never partway through the solution process.
5. Applications to physical systems
Ordinary differential equations describe an enormous range of physical systems using the same handful of techniques above: simple harmonic and damped oscillators (Physics Chapter III), RLC electrical circuits (Physics Chapter I of Paper II), Newton’s law of cooling, and population growth/decay models — recognising that these superficially different physical situations all reduce to the same small number of standard ODE forms is one of the most valuable unifying insights this chapter provides.
Key Takeaways
- Identifying an equation’s type (separable, linear, exact) determines its solution method; the integrating factor systematically solves any linear first-order equation.
- Constant-coefficient higher-order equations reduce to an algebraic characteristic equation whose root type (real distinct, repeated, complex) determines the solution’s qualitative form.
- The general solution of a non-homogeneous equation is always complementary function plus particular integral; both parts are necessary.
- Initial conditions (single point) and boundary conditions (multiple points) both fix the arbitrary constants in a general solution, but apply in different problem types and must be applied only to the fully combined general solution.
- The same standard ODE techniques recur across superficially different physical applications — oscillators, circuits, cooling, population models — making this chapter a genuine unifying tool across the wider syllabus.