# How to | Solve a Differential Equation

*Mathematica*'*s* differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms without needing preprocessing by the user.

Use DSolve to solve the differential equation for with independent variable :

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The solution given by DSolve is a list of lists of rules. The outermost list encompasses all the solutions available, and each smaller list is a particular solution.

If you want to use a solution as a function, first assign the rule to something, in this case, :

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Now, use Part to take the first part of the solution using the short form . Replace using (the short form of ReplaceAll) and then use to define the function :

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Now, evaluates like any normal function:

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To specify initial conditions, enclose the equation and the initial conditions ( and ) in a list:

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If not enough initial conditions are given, constants C[n] are returned:

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To indicate which functions should be solved for, use a second list:

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Here the solutions are not elementary functions:

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You can use DSolve, , Table, and Plot together to graph the solutions to an underspecified differential equation for various values of the constant.

First, solve the differential equation using DSolve and set the result to :

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Use , , and Part to define a function using :

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Define a table of functions for integer values of C[2] between 1 and 10:

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Use Plot to plot the table over the range :

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