![]() ![]() In no way is this description to be considered complete. Precisely, providing approximations to ordinary differential equations (ODEs for short). This section provides an idea of the basic methods behind numerical methods of solving or, more Return to the main page for the course APMA0346 Return to the main page for the course APMA0340 Return to the main page for the course APMA0330 Return to Mathematica tutorial for the fourth course APMA0360 Return to Mathematica tutorial for the second course APMA0340 ![]() Return to Mathematica tutorial for the first course APMA0330 Return to computing page for the fourth course APMA0360 Return to computing page for the second course APMA0340 Return to computing page for the first course APMA0330 Laplace transform of discontinuous functions.Series Solutions near a regular singular point.Series solutions near ordinary singular point.Series Solutions for the Second Order Equations.Picard iterations for the second order ODEs.Part IV: Second and Higher Order Differential Equations.Numerical solution using DSolve and NDSolve. ![]() Part III: Numerical Methods and Applications.Equations reducible to the separable equations.X t = &ExponentialE − t, y t = − &ExponentialE − t + &ExponentialE −1įor detailed information on the dsolve command, see dsolve/details. Solve the system of ODEs subject to the initial conditions ics. X t = c_2 &ExponentialE − t, y t = − c_2 &ExponentialE − t + c_1 If the unknowns are not specified, all differentiated indeterminate functions in the system are treated as the unknowns of the problem. Sys_ode ≔ &DifferentialD &DifferentialD t y t = x t, &DifferentialD &DifferentialD t x t = − x t Sys_ode ≔ &DifferentialD &DifferentialD t y t = x t, &DifferentialD &DifferentialD t x t = − x t Odetest series_sol, ode, ics, series Series_sol ≔ dsolve ode, ics, y x, series Test whether the ODE solution satisfies the ODE and the initial conditions (see odetest ).įind a series solution for the same problem. Sol ≔ dsolve ode, ics, y x, method = laplace Y x = 3 &ExponentialE 2 x 4 + 3 &ExponentialE − 2 x 4 − 1 2Ĭompute the solution using the Laplace transform method. Solve ode subject to the initial conditions ics. Y x = &ExponentialE 2 x c_2 + &ExponentialE − 2 x c_1 − 1 2 Ode ≔ &DifferentialD 2 &DifferentialD x 2 y x = 2 y x + 1 Ode ≔ &DifferentialD 2 &DifferentialD x 2 y x = 2 y x + 1 To define a derivative, use the diff command or one of the notations explained in Derivative Notation. For more information, see dsolve and worksheet/interactive/dsolve. Using the assistant, you can compute numeric and exact solutions and plot the solutions. ![]() The ODE Analyzer Assistant is a point-and-click interface to the ODE solver routines. Computing numerical (see dsolve/numeric ) or series solutions (see dsolve/series ) for ODEs or systems of ODEs. Computing solutions using integral transforms (Laplace and Fourier). Computing formal solution for a linear ODE with polynomial coefficients. Computing formal power series solutions for a linear ODE with polynomial coefficients. Solving ODEs or a system of them with given initial conditions (boundary value problems). Computing closed form solutions for a single ODE (see dsolve/ODE ) or a system of ODEs, possibly including anti-commutative variables (see dsolve/system ). (See the Examples section.)Īs a general ODE solver, dsolve handles different types of ODE problems. (optional) depends on the type of ODE problem and method used, for example, series or method=laplace. , where are constants with respect to the independent variable Initial conditions of the form y(a)=b, D(y)(c)=d. Ordinary differential equation, or a set or list of ODEsĪny indeterminate function of one variable, or a set or list of them, representing the unknowns of the ODE problem Solve ordinary differential equations (ODEs) ![]()
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