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DSolve

  • DSolve[ eqn , y , x ] solves a differential equation for the function y, with independent variable¬†x.
  • DSolve[ , , ... , , , ... , x ] solves a list of differential equations.
  • DSolve[ eqn , y , , , ... ] solves a partial differential equation.
  • DSolve[ eqn , y [ x ], x ] gives solutions for y [ x ] rather than for the function y itself.
  • Example: DSolve[y'[x] == 2 a x, y[x], x].
  • Differential equations must be stated in terms of derivatives such as y '[ x ], obtained with D, not total derivatives obtained with Dt.
  • DSolve generates constants of integration indexed by successive integers. The option DSolveConstants specifies the function to apply to each index. The default is DSolveConstants->C, which yields constants of integration C[1], C[2], ... .
  • DSolveConstants->(Module[{C}, C]&) guarantees that the constants of integration are unique, even across different invocations of DSolve.
  • For partial differential equations, DSolve generates arbitrary functions C[ n ][... ].
  • Boundary conditions can be specified by giving equations such as y'[0] == b.
  • Solutions given by DSolve sometimes include integrals that cannot be carried out explicitly by Integrate. Dummy variables with local names 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 solve also many linear equations up to second order with non-constant coefficients.
  • DSolve includes general procedures that handle a large fraction of 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.
  • See the Mathematica book: Section 1.5.8,¬†Section 3.5.10.
  • See also Implementation NotesA.9.55.13MainBookLinkOldButtonDataA.9.55.13.
  • See also: NDSolve, Solve.
  • Related package: Calculus`VariationalMethods`, Calculus`VectorAnalysis`.

    Further Examples

    Here is the solution to a first-order differential equation.

    In[1]:=

    Out[1]=

    This gives the solution using C[1] and C[2] as the constants of integration.

    In[2]:=

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    This solves two simultaneous differential equations.

    In[3]:=

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    You can add constraints and boundary conditions for differential equations by explicitly giving additional equations such as y[0] == 5 as in this example.

    In[4]:=

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    This solves a first-order differential equation, specifying that the integration constant is K.

    In[5]:=

    Out[5]=