# Wolfram Language & System 10.0 (2014)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# 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 , 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. »

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Solve a differential equation:

 Out[1]=

Include a boundary condition:

 Out[2]=

Get a "pure function" solution for :

 Out[1]=

Substitute the solution into an expression:

 Out[2]=