# Linear Second-Order Equations with Solutions Involving Special Functions

DSolve can find solutions for most of the standard linear second-order ODEs that occur in applied mathematics.

Here is the solution for

*Airy's equation*.

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Here is a plot that shows the oscillatory behavior of the Airy functions for large negative values of

.

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The solution to this equation is given in terms of the derivatives of the Airy functions,

AiryAiPrime and

AiryBiPrime.

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Here is the solution for

*Bessel's equation* with

. Note that the solution is given in terms of Bessel functions of the first kind,

BesselJ, as well as those of the second kind,

BesselY.

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Here is a plot of the

BesselJ functions for specific values of

.

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Here is the general solution for

*Legendre's equation *with

.

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These special functions can be expressed in terms of elementary functions for certain values of their parameters. *Mathematica* performs this conversion automatically wherever it is possible.

These are some of these expressions that are automatically converted.

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As a result of these conversions, the solutions of certain ODEs can be partially expressed in terms of elementary functions. Hermite's equation is one such ODE.

Here is the solution for Hermite's equation with arbitrary

.

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With

set to 5, the solution is given in terms of polynomials, exponentials, and

Erfi.

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