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dsolve

solve ordinary differential equations (ODEs)

 

Calling Sequence

Parameters

Description

Examples

Details

Calling Sequence

dsolve(ODE)

dsolve(ODE, y(x), options)

dsolve({ODE, ICs}, y(x), options)

Parameters

ODE

-

ordinary differential equation, or a set or list of ODEs

y(x)

-

any indeterminate function of one variable, or a set or list of them, representing the unknowns of the ODE problem

ICs

-

initial conditions of the form y(a)=b, D(y)(c)=d, ..., where {a, b, c, d} are constants with respect to the independent variable

options

-

(optional) depends on the type of ODE problem and method used, for example, series or method=laplace. (See the Examples section.)

Description

• 

As a general ODE solver, dsolve handles different types of ODE problems. These include the following.

  

- Computing closed form solutions for a single ODE (see dsolve/ODE) or a system of ODEs, possibly including anti-commutative variables (see dsolve/system).

  

- Solving ODEs or a system of them with given initial conditions (boundary value problems). See dsolve/ICs.

  

- Computing formal power series solutions for a linear ODE with polynomial coefficients. See dsolve/formal_series.

  

- Computing formal solution for a linear ODE with polynomial coefficients. See dsolve/formal_solution.

  

- Computing solutions using integral transforms (Laplace and Fourier). See dsolve/integral_transform.

  

- Computing numerical (see dsolve/numeric) or series solutions (see dsolve/series) for ODEs or systems of ODEs.

• 

The ODE Analyzer Assistant is a point-and-click interface to the ODE solver routines. Using the assistant, you can compute numeric and exact solutions and plot the solutions. For more information, see dsolve[interactive] and worksheet/interactive/dsolve.

• 

To define a derivative, use the diff command or one of the notations explained in Derivative Notation.

Examples

Solving an ODE

  

Define a simple ODE.

odediffyx,x,x=2yx+1

odeⅆ2ⅆx2yx=2yx+1

(1)
  

Solve the ODE, ode.

dsolveode

yx=ⅇ2x_C2+ⅇ2x_C112

(2)
  

Define initial conditions.

icsy0=1,Dy0=0

icsy0=1,Dy0=0

(3)
  

Solve ode subject to the initial conditions ics.

dsolveics,ode

yx=3ⅇ2x4+3ⅇ2x412

(4)

Laplace Transform Method

  

Compute the solution using the Laplace transform method.

soldsolveics,ode,yx,method=laplace

solyx=12+3cosh2x2

(5)
  

Test whether the ODE solution satisfies the ODE and the initial conditions (see odetest).

odetestsol,ode,ics

0,0,0

(6)

Computing a Series Solution

  

Find a series solution for the same problem.

series_soldsolveics,ode,yx,series

series_solyx=1+32x2+14x4+Ox6

(7)

odetestseries_sol,ode,ics,series

0,0,0

(8)

Solving an ODE System

  

Define a system of ODEs.

sys_odediffyt,t=xt,diffxt,t=xt

sys_odeⅆⅆtyt=xt,ⅆⅆtxt=xt

(9)
  

If the unknowns are not specified, all differentiated indeterminate functions in the system are treated as the unknowns of the problem.

dsolvesys_ode

xt=_C2ⅇt,yt=_C2ⅇt+_C1

(10)
  

Define initial conditions.

icsx0=1,y1=0

icsx0=1,y1=0

(11)
  

Solve the system of ODEs subject to the initial conditions ics.

dsolvesys_ode,ics

xt=ⅇt,yt=ⅇt+ⅇ−1

(12)

Details

• 

For detailed information on the dsolve command, see dsolve/details.

See Also

DEtools

diff

dsolve/algorithms

dsolve/details

dsolve/education

dsolve/formal_series

dsolve/formal_solution

dsolve/hypergeometric

dsolve/ICs

dsolve/integrating_factors

dsolve/integrating_factors_for_LODEs

dsolve/inttrans

dsolve/Lie

dsolve/linear

dsolve/numeric

dsolve/piecewise

dsolve/references

dsolve/series

dsolve/system

ODE Analyzer Assistant

odeadvisor

odeadvisor/types

PDEtools

pdsolve