Solving First-Order Differential Equations
Differential equations describe how quantities change over time or space. A first-order ordinary differential equation (ODE) involves a function and its first derivative. These equations model everything from population growth and radioactive decay to circuit analysis and heat transfer.
Separable equations are the simplest type. They have the form dy/dx equals g(x) times h(y). You separate variables by dividing both sides by h(y) and multiplying by dx, giving dy/h(y) equals g(x) dx. Integrate both sides to find the solution. For example, dy/dx equals x/y separates to y dy equals x dx, integrating to y^2/2 equals x^2/2 plus C, or y^2 equals x^2 plus C.
Linear first-order equations have the standard form dy/dx plus P(x)y equals Q(x). The integrating factor mu(x) equals e raised to the integral of P(x) dx. Multiplying both sides by mu(x) gives d/dx[mu(x)y] equals mu(x)Q(x), so y equals one over mu(x) times the integral of mu(x)Q(x) dx.
Exact equations come from the total differential of a two-variable function. If M(x,y)dx plus N(x,y)dy equals zero and the partial derivative of M with respect to y equals the partial derivative of N with respect to x, the equation is exact. The solution is psi(x,y) equals C, where psi_x equals M and psi_y equals N.
Euler's method provides numerical solutions when analytical methods fail. Starting from an initial condition, you step forward using the formula y_(n+1) equals y_n plus h times f(x_n, y_n), where h is the step size. The differential equation solver on this site implements both Euler's method and the fourth-order Runge-Kutta method (RK4), which is far more accurate.
Applications are everywhere. Newton's law of cooling says dT/dt equals negative k(T minus T_env), giving T(t) equals T_env plus (T_0 minus T_env) times e^(-kt). Population models use dP/dt equals rP(1 minus P/K) for logistic growth. Mixing problems involve rates of substance entering and leaving a tank.
The slope field generator visualizes the direction field of a first-order ODE, giving geometric intuition about solution behavior without solving the equation analytically.
Second-order linear ODEs with constant coefficients follow a characteristic equation. For ay'' plus by' plus cy equals 0, try y equals e^(rt), getting ar^2 plus br plus c equals 0. The roots determine the form: two distinct real roots give C1 e^(r1 t) plus C2 e^(r2 t); repeated roots give (C1 plus C2 t) e^(rt); complex roots give e^(alpha t)(C1 cos beta t plus C2 sin beta t).