DSolve

DSolve[eqn,y,x]
solves a differential equation for the function y, with independent variable x.

DSolve[eqn,y,{x,xmin,xmax}]
solves a differential equation for x between and .

DSolve[{eqn1,eqn2,},{y1,y2,},]
solves a list of differential equations.

DSolve[eqn,y,{x1,x2,}]
solves a partial differential equation.

Details and OptionsDetails and Options

  • DSolve[eqn,y[x],x] gives solutions for rather than for the function y itself.
  • Differential equations must be stated in terms of derivatives such as , obtained with D, not total derivatives obtained with Dt.
  • Boundary conditions can be specified by giving equations such as .
  • The list of equations given to DSolve can include algebraic ones that do not involve derivatives.
  • DSolve can also solve delay differential equations.
  • In delay differential equations, initial history functions are given in the form , where is in general a function of x.
  • WhenEvent[event,action] may be included in the equations eqn to specify an action that occurs when event becomes True.
  • DSolve[eqn,y,{x,xmin,xmax}] gives a solution valid for x between and .
  • DSolve can generate constants of integration indexed by successive integers. The option GeneratedParameters specifies the function to apply to each index. The default is GeneratedParameters->C, which yields constants of integration C[1], C[2], . »
  • GeneratedParameters->(Module[{C},C]&) guarantees that the constants of integration are unique, even across different invocations of DSolve.
  • For partial differential equations, DSolve typically generates arbitrary functions C[n][]. »
  • Solutions given by DSolve sometimes include integrals that cannot be carried out explicitly by Integrate. Variables , , , are used in such integrals.
  • DSolve sometimes gives implicit solutions in terms of Solve. »
  • DSolve can solve linear ordinary differential equations of any order with constant coefficients. It can also solve many linear equations up to second order with nonconstant coefficients.
  • DSolve includes general procedures that handle almost all the nonlinear ordinary differential equations whose solutions are given in standard reference books such as Kamke.
  • DSolve can find general solutions for linear and weakly nonlinear partial differential equations. Truly nonlinear partial differential equations usually admit no general solutions.
  • DSolve can handle not only pure differential equations but also differentialalgebraic equations. »
Introduced in 1991
(2.0)
| Updated in 2014
(10.0)