#### SymmetricPower

#### UserFunction SymmetricPower

`SymmetricPower(u:LODE(C,y,x),n:Z)`

The first argument $u$ has the type $LODE$ which abbreviates *LinearOrdinarDifferentialEquation*.

The meaning of its parameters is as follows.

$C:$ Type of coefficients.

$y:$ Dependent variable.

$x:$ Independent variable.

**Specification.** The $n$-th symmetric power of the first argument is returned.

**Examples.** The input for the examples is given in *Reduce* algebraic mode syntax. The output is returned in a separate window.

`deq:=Df(y,x,2)-x*y**2+(3/(16*x**2)-1/x)*y;`

`s2:=SymmetricPower(deq,2);`

$y''-\frac{x-\frac{3}{16}}{x^{2}}{y} = 0$

The $2^{nd}$ symmetric power:

$y'''-\frac{16x-3}{4x^{2}}y'+\frac{2x-\frac{3}{4}}{x^{3}}y= 0$

**Related Functions.** SymmetricProduct.

**References.**

M. Singer, * Solving Homogeneous Linear Differential Equations in Terms of Second Order Linear Differential Equations*,

American Journal of Mathematics 107, 663-696.

F. Schwarz, *Algorithmic Lie Theory for Solving Ordinary Differential Equations*, CRC Press, 2007.

##### User Interface Functions