Differential Equations

Differential equations relate a function to its derivatives, describing how systems evolve over time. This topic covers separable and first-order linear equations, second-order equations with constant coefficients, homogeneous and exact equations, Bernoulli equations, and systems of ODEs.

Differential equations model population dynamics, heat transfer, electrical circuits, mechanical vibrations, and fluid flow. They are the primary language for describing change in engineering and the natural sciences.

Practice Tips

• 1Classify the equation type first (separable, linear, exact, Bernoulli) before choosing a solution method, since each type has its own standard approach.
• 2For second-order linear equations with constant coefficients, start by solving the characteristic equation and note whether roots are real, repeated, or complex.
• 3Always check your solution by substituting it back into the original equation and verifying that initial conditions are satisfied.