Differential Equation Calculator
Solve first-order ODEs: exponential, logistic, and separable forms.
Differential Equation Calculator: Solving ODEs
A differential equation relates a function to one or more of its derivatives. They model how quantities change over time or space, making them fundamental to physics, engineering, biology, and economics. Our calculator solves the most common first-order ordinary differential equations (ODEs), providing both the general symbolic solution and a particular numeric solution.
Exponential Growth and Decay: dy/dx = ky
The most important first-order ODE. When k > 0, the solution describes exponential growth — bacterial populations, compound interest, radioactive production. When k < 0, it models decay — radioactive half-life, drug elimination, Newtonian cooling. The solution is always y(x) = y0 · ekx.
Logistic Growth
The logistic ODE dy/dx = ky(1 − y/L) captures bounded growth where carrying capacity L prevents unlimited expansion. As y → L, growth rate approaches zero producing the characteristic S-shaped sigmoid curve used in ecology, epidemiology, and market saturation modeling.
Separable Equations
A separable ODE can be written as dy/dx = f(x)/g(y), allowing all y terms to shift to one side and all x terms to the other before integrating independently. This tool handles linear separable cases computing implicit solutions and evaluating y numerically.