#### 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);

The Differential Equation:

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

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