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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]:= 
Out[2]= 
This solves two simultaneous differential equations.
In[3]:= 
Out[3]= 
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]:= 
Out[4]= 
This solves a first-order differential equation, specifying that the integration constant is K.
In[5]:= 
Out[5]= 
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