Question

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution x^3 (y''') + 6x^2 (y'') + 4xy' - 4y =0; x, x^-2, x^-2 (lnx), (0,infinity)?

Answer 1

The functions $f_1(x)=x,f_2(x)=x^{-2},f_3(x)=x^{-2}lnx$ are linearly independent if the Wronskian:

$W=\begin{vmatrix} f_1 & f_2&f_3 \\ f'_1 & f'_2&f'_3\\ f''_1 & f''_2&f''_3\\ \end{vmatrix}\ne 0$

$W=\begin{vmatrix} x & x^{-2}&x^{-2}lnx \\ 1 & -2x^{-3}&-2x^{-3}lnx+x^{-3}\\ 0 & 6x^{-4}&6x^{-4}lnx-2x^{-4}-3x^{-4}\\ \end{vmatrix}=$

$=x\begin{vmatrix} -2x^{-3} & -2x^{-3}lnx+x^{-3} \\ 6x^{-4} & 6x^{-4}lnx-5x^{-4} \end{vmatrix}-\begin{vmatrix} x^{-2} & x^{-2}lnx \\ 6x^{-4} & 6x^{-4}lnx-5x^{-4} \end{vmatrix}=$

$=x(-12x^{-7}lnx+10x^{-7}+12x^{-7}lnx-6x^{-7})-$

$-(6x^{-6}lnx-5x^{-6}-6x^{-6}lnx)=x(4x^{-7})+5x^{-6}=x^{-6}\ne 0$

So, the given functions form a fundamental set of solutions of the differential equation on the indicated interval.

The general solution:

$y(x)=c_1x+c_2x^{-2}+c_3x^{-2}lnx$