Tools
Step-by-step solvers
Enter your own coefficients or functions and every solver shows the full solution one step at a time, just as you would write it on paper. Each tool is linked to the section that explains the method.
Chapter 1: Differential Equations and Their Solutions
Slope Field (Direction Field) PlotterPlot the slope field of y′ = f(x, y) and click to draw solution curves.Differential Equation Solution CheckerCheck step by step whether a function solves a differential equation.Existence and Uniqueness Theorem CheckerTest the hypotheses of the existence and uniqueness theorem at a point.
Chapter 2: First-Order Equations with Exact Solutions
Exact Differential Equation SolverExactness test, integrating factors μ(x) or μ(y), and the implicit solution F(x, y) = C.Separable Differential Equation SolverSeparate, integrate, recover equilibrium solutions and solve for y.Homogeneous Differential Equation Solver (y = vx)Solve y′ = (ax + by)/(cx + dy) with the substitution y = vx.Linear-fractional equation solvery′ = (a₁x + b₁y + c₁)/(a₂x + b₂y + c₂) by translation or z = a₁x + b₁y.First-Order Linear Differential Equation SolverIntegrating factor method for y′ + P(x)y = Q(x), with initial conditions.Bernoulli Equation SolverReduce y′ + Py = Qyⁿ to a linear equation with v = y^(1−n).
Chapter 3: Applications of First-Order Equations
Orthogonal Trajectories CalculatorDifferential equation of a family, the perpendicular family, and a plot of both.Falling Body with Air Resistance CalculatorVelocity, distance and terminal velocity for linear drag.Exponential Growth and Decay CalculatorRate constant from data, doubling time or half-life, predictions.Newton's Law of Cooling CalculatorCooling constant from one reading, temperature at any time, time to reach a temperature.Mixing Tank Problem SolverAmount of solute for equal or unequal inflow and outflow rates.Logistic Growth CalculatorLogistic solution, inflection point and predictions.
Chapter 4: Higher-Order Linear Equations: Explicit Methods
Wronskian CalculatorWronskian of two or three functions and a verdict on linear independence.Constant-Coefficient ODE Solver (Characteristic Equation)Characteristic equation, roots (with synthetic division), general solution and IVPs, up to order 6.Undetermined Coefficients CalculatorTrial form with the modification rule, coefficient matching and the general solution.Variation of Parameters CalculatorWronskian, u₁′ and u₂′, integration and the particular solution.Cauchy–Euler Equation SolverSubstitution x = eᵗ, indicial equation, and nonhomogeneous terms in powers of x and ln x.
Chapter 5: Applications of Second-Order Linear Equations with Constant Coefficients
Mass–Spring–Damper CalculatorFree and forced vibrations: damping type, solution, amplitude–phase form, steady state, graph.Resonance CalculatorAmplitude response curves and the resonance frequency.RLC Circuit Differential Equation SolverCharge and current in a series RLC circuit, transient and steady state.
Chapter 6: Series Solutions of Linear Differential Equations
Power Series Solution CalculatorRecurrence relation and series solutions about an ordinary point.Frobenius Method CalculatorIndicial equation, the three cases, recurrence and Frobenius series.
Chapter 7: Systems of Linear Differential Equations
2×2 Linear System Solver with Phase PortraitEigenvalues, eigenvectors, general solution, IVP and phase portrait.3×3 Linear System Solver (Eigenvalue Method)Eigenvalues, eigenvectors and generalized eigenvectors for x′ = Ax.
Chapter 8: Approximate Methods for First-Order Equations
Taylor Series Method CalculatorSuccessive derivatives at the initial point and the Taylor polynomial.Picard Iteration CalculatorSuccessive approximations φ₁, φ₂, … with exact polynomial arithmetic.Euler, Runge–Kutta (RK4) and ABAM CalculatorStep-by-step tables with every intermediate value and errors against the exact solution.Runge–Kutta Solver for SystemsEuler and RK4 for two coupled equations (and second-order equations).
Chapter 9: The Laplace Transform
Laplace Transform CalculatorTransforms term by term with the rule used for each.Inverse Laplace Transform CalculatorPartial fractions and term-by-term inversion.Convolution Calculator (Laplace Transform)f * g through the convolution theorem, with a numerical check.Laplace Transform IVP SolverIVPs with formula, step/pulse or impulse forcing.Laplace Transform Solver for SystemsTwo linear equations with initial conditions via Cramer's rule in s.