# 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.

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.

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

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].

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.

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

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.

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.

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

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

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

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.