# Setting Up the Problem

The first argument given to DSolve is the differential equation, the second argument is the unknown function, and the last argument identifies the independent variable.

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The output of DSolve is a list of solutions for the differential equation. The extra list is required since some equations have multiple solutions. Here, since the equation is of order 1 and is linear, there is only one solution: y[x]->+^{-5 x} C[1]. The solution has an undetermined constant C[1] because no initial condition was specified. The solution can be extracted from the list of solutions using a part specification.

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This form of the solution is useful for finding itself, but not for finding derivatives of or the value of at a point.

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If the solution will be used in further work, it is best to specify the unknown function using rather than . This gives the solution using pure functions of the type Function[x,expr].

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When the solution is in the form of pure functions, expressions can be found for derivatives of and for the values of at specific points.

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When a problem has multiple solutions, you can pick out individual solutions from the solution list or you can work directly with the list.

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To solve a system of equations, the first argument to DSolve must be a list of the equations and the second argument must be a list of the unknown functions.

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Each solution to the system is a list of replacement rules for the unknown functions. The expressions for the unknown functions can be extracted as in previous examples.

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If initial conditions are prescribed for the problem, some or all of the undetermined constants can be eliminated.

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For a partial differential equation, the third argument to DSolve is a list of the independent variables for the equation.

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A differential-algebraic equation is specified in the same way as a system of ordinary differential equations.

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Note that it is not always possible to give the solutions for a problem in explicit form. In this case, the solution is given using an unevaluated Solve object or using InverseFunction.

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