Differential Equations for Engineering
Systematic guide to ODEs — from separable equations to Laplace transforms. Click a topic to explore.
Separable, linear, Bernoulli, and substitution methods
A first-order ODE involves only y and y'. The solution method depends entirely on the equation's form. The three main types — separable, linear, and Bernoulli — each have a systematic algorithm. Identify the type first, then apply the corresponding method.
Before solving, classify the ODE. An ordinary differential equation (ODE) involves one independent variable and its derivatives. The order is the highest derivative present.
A general solution to an nth-order ODE contains n arbitrary constants. It represents a family of infinitely many solution curves. Initial conditions (ICs) pin down one particular solution.
Apply the initial condition y(x₀) = y₀ to the general solution to determine the arbitrary constant(s). The result is a unique function satisfying both the ODE and the IC.
An ODE is linear if it has the form aₙy⁽ⁿ⁾ + ⋯ + a₁y' + a₀y = f(x). Nonlinear terms include y², (y')², sin(y), or y·y'. First-order means the highest derivative is y'.
Slope / Direction Field
Click anywhere on the field to draw a solution curve (Euler method)
Q1.The integrating factor μ(x) for the linear ODE dy/dx + P(x)y = Q(x) is: