Equation Solving

Solve algebraic and differential equations

You can solve algebraic equations, differential equations, and differential algebraic equations (DAEs).

Solve algebraic equations to get either exact analytic solutions or high-precision numeric solutions. For analytic solutions, use solve, and for numerical solutions, use vpasolve. For solving linear equations, use linsolve. These solver functions have the flexibility to handle complicated problems. See Troubleshoot Equation Solutions from solve Function.

Solve differential equations by using dsolve. Create these differential equations by using symbolic functions. See Create Symbolic Functions.

Solve differential algebraic equations (DAEs) by first reducing their differential index to 1 or 0 using Symbolic Math Toolbox™ functions, and then using MATLAB^® solvers, such as ode15i, ode15s, or ode23t. See Solve Differential Algebraic Equations (DAEs).

Live Editor Tasks

Solve Symbolic Equation

Find analytic solutions of symbolic equations in Live Editor (Since R2020a)

Functions

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Linear and Nonlinear Equations and Systems

`equationsToMatrix`	Convert linear equations to matrix form
`eliminate`	Eliminate variables from rational equations
`finverse`	Functional inverse
`isolate`	Isolate variable or expression in equation
`linsolve`	Solve symbolic linear equations in matrix form
`poles`	Poles of expression or function
`solve`	Equations and systems solver
`vpasolve`	Solve symbolic equations numerically

Ordinary Differential Equations (ODEs)

`dsolve`	Solve system of differential equations
`massMatrixForm`	Extract mass matrix and right side of semilinear system of differential algebraic equations
`odeFunction`	Convert symbolic expressions to function handle for ODE solvers
`odeToVectorField`	Reduce order of differential equations to first-order

Differential Algebraic Equations (DAEs)

`daeFunction`	Convert system of differential algebraic equations to MATLAB function handle suitable for `ode15i`
`decic`	Find consistent initial conditions for first-order implicit ODE system with algebraic constraints
`findDecoupledBlocks`	Search for decoupled blocks in systems of equations
`incidenceMatrix`	Find incidence matrix of system of equations
`isLowIndexDAE`	Check if differential index of system of equations is lower than 2
`massMatrixForm`	Extract mass matrix and right side of semilinear system of differential algebraic equations
`odeFunction`	Convert symbolic expressions to function handle for ODE solvers
`reduceDAEIndex`	Convert system of first-order differential algebraic equations to equivalent system of differential index 1
`reduceDAEToODE`	Convert system of first-order semilinear differential algebraic equations to equivalent system of differential index 0
`reduceDifferentialOrder`	Reduce system of higher-order differential equations to equivalent system of first-order differential equations
`reduceRedundancies`	Simplify system of first-order differential algebraic equations by eliminating redundant equations and variables

Partial Differential Equations

`pdeCoefficients`	Extract coefficients of partial differential equation (Since R2021a)
`pdeCoefficientsToDouble`	Convert symbolic PDE coefficients to `double` format (Since R2021a)

Topics

Solve Algebraic Equations

Solve equations analytically, return full solutions, and visualize results.
Solve Differential Equation

Solve a differential equation analytically by using the dsolve function, with or without initial conditions.
Solve Differential Algebraic Equations (DAEs)

This example show how to solve differential algebraic equations (DAEs) by using MATLAB® and Symbolic Math Toolbox™.

Related Information

Featured Examples

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Validate Simulink Model Using Symbolic Math Toolbox

Model a bouncing ball, which is a classical hybrid dynamic system. This model includes both continuous dynamics and discrete transitions. It uses the Symbolic Math Toolbox™ to help explain some of the theory behind ODE solving in the Simulation of a Bouncing Ball.

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Solve polynomial equations and systems of equations, and work with the results using Symbolic Math Toolbox™.

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The Black–Scholes Formula for Call Option Price

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Heating of Finite Slab

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Finite Element Formulation for Timoshenko Beam Problem

Apply the finite element method (FEM) to solve a Timoshenko beam problem, using both linear and quadratic basis functions for analysis. The Timoshenko beam theory is a 1-D problem that reduces the complex 3-D problem of beam deformation to a set of 1-D differential equations along the length of the beam. In contrast to the Euler–Bernoulli beam theory, which does not consider shear deformation, the Timoshenko beam theory accounts for both shear deformation and rotational bending effects. The Timoshenko beam theory is generally more accurate for short, thick beams where shear deformation cannot be neglected. Using a cantilever beam and a beam fixed at both ends, this example discusses the analytical solutions of the Timoshenko beam problem and compares them with the FEM solutions.

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