Question

Find a fundamental set of solutions {_{2} " alt="y_{1},y_{2} " align="absmiddle" class="latex-formula">}of the equation $x^{2} \frac{d^{2}y}{dx^{2}} - 13x \frac{dy}{dx} + 49y=0$ in the interval x>0

Answer 1

Another Cauchy-Euler ODE.

$y=x^r$
$\implies r(r-1)-13r+49=r^2-14r+49=(r-7)^2=0$
$\implies r=7$
$\implies y=C_1x^7+C_2x^7\ln x$

where $y_1=x^7$ and $y_2=x^7\ln x$.

To verify that these solutions are linearly independent, check the Wronskian:

$W(y_1,y_2)=\begin{vmatrix}x^7&x^7\ln x\\7x^6&x^6(7\ln x+1)\end{vmatrix}=x^{13}$

and so the solutions are indeed independent.