Equation Solving
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 Choose an Approach for Solving Equations Using 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 |
Functions
Topics
- Solve Algebraic Equations
Solve equations analytically, return full solutions, and visualize results.
- Solve Differential Equation
Solve a differential equation analytically by using the
dsolvefunction, 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™.
- Substitute Values into Symbolic Expressions Using subs
Substitute values into symbolic expressions for various tasks such as evaluation, simplification, and verification.
Featured Examples
The Physics of the Damped Harmonic Oscillator
Explores the physics of the damped harmonic oscillator by solving the equations of motion in the case of no driving forces. This example investigates the cases of underdamping, overdamping, and critical damping.
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.
Derive and Apply Inverse Kinematics to Two-Link Robot Arm
Derive and apply inverse kinematics to a two-link robot arm by using MATLAB and Symbolic Math Toolbox.
Gauss-Laguerre Quadrature Evaluation Points and Weights
Solve polynomial equations and systems of equations, and work with the results using Symbolic Math Toolbox™.
Simulate a Stochastic Process Using the Feynman–Kac Formula
Obtains the partial differential equation that describes the expected final price of an asset whose price is a stochastic process given by a stochastic differential equation.
The Black–Scholes Formula for Call Option Price
Calculate the call option price using the Black–Scholes formula. This example uses vpasolve to numerically solve the problems of finding the spot price and implied volatility from the Black–Scholes formula.
Heating of Finite Slab
Find the temperature distribution of a one-dimensional finite slab by solving the differential equation using the method of separation of variables.
Derive Equations of Motion and Simulate Cart-Pole System
Derive the equations of motion for the cart-pole system using Symbolic Math Toolbox™ and then simulate the cart-pole system using the ode45 solver. In the later sections of the example, you explore how to derive the equations in other forms that can be used to numerically simulate the system (and validate the results) using different tools, such as Simulink®, Simscape™ Multibody™, and Robotics System Toolbox™.
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.
Analyze Transfer Function of T-Coil Circuit
Analyze the transfer function of a T-coil circuit using Symbolic Math Toolbox™ and Control System Toolbox™. In this example, you define symbolic equations, solve for the transfer function, and analyze the stability, frequency response, and step response of the circuit.
Teaching Resources
Beam Bending and Deflection
Learn the concepts of solving beam bending and deflection problems symbolically, and then visualize the results.
Applied Ordinary Differential Equations
Learn how to classify ODEs, and methods of solution including separation of variables and integrating factors.
Numerical Methods with Applications
Learn methods for interpolation, numerical integration and derivation, and finite difference methods for differential equations.
