Ordinary differential equations
Also known as: ODE
An ordinary differential equation (ODE) is an equation relating a function of a single variable to its derivatives. Solving an ODE means finding the function or family of functions that satisfies the equation.
An ordinary differential equation (ODE) is an equation that involves an unknown function of one independent variable and its derivatives — for example, dy/dx + 2y = 0. "Ordinary" distinguishes it from a partial differential equation, which involves partial derivatives of a function of several variables. The order of an ODE is the highest derivative that appears: y′ + 2y = 0 is first order, while y″ + 3y′ + 2y = 0 is second order.
Solving an ODE means finding the function that satisfies it. The general solution contains arbitrary constants — one per order — representing an entire family of solutions. Applying initial conditions pins down the constants and produces a particular solution. For instance, dy/dx = 2y has general solution y = Ce²ˣ; the initial condition y(0) = 3 selects the particular solution y = 3e²ˣ.
Standard solution methods depend on the equation's form. First-order separable equations are solved by separation of variables; linear first-order equations use an integrating factor; and linear second-order equations with constant coefficients are solved with the characteristic equation, whose roots determine exponential, repeated, or sinusoidal solution forms. ODEs are the standard language for modeling change — exponential growth and decay, mixing problems, circuits, and damped vibrations all reduce to ODEs.
The FE Mechanical exam includes ordinary differential equations in its mathematics section. Be ready to classify an ODE by order and linearity, solve separable and linear first-order equations, and solve second-order constant-coefficient equations using the characteristic equation.
Key takeaways
- An ODE relates a function of a single variable to its derivatives; its order is the highest derivative present.
- The general solution contains one arbitrary constant per order; initial conditions produce a particular solution.
- Common techniques include separation of variables, integrating factors, and the characteristic equation for constant-coefficient equations.
- ODEs model real processes such as exponential growth and decay, mixing problems, and vibrations.
- The FE Mechanical exam tests classifying ODEs and solving first- and second-order equations.