Question

Find the general solution of the equation:

𝑥^2𝑦" − 9𝑥𝑦′ + 25𝑦 = 0

Answer 1

$x=e^u$

$a_0\lambda(\lambda-1)(\lambda-2)...(\lambda-n+2)...+a_{n-2}\lambda(\lambda-1)+a_{n-1}\lambda+a_n=0$

$(\lambda-1)\lambda-9\lambda+25=0$

$\lambda^2-10\lambda+25=0$

$\lambda_{1,2}=5$

k=2

$y=(C_1u+C)e^{5u}$

$y=C_1x^5ln(x)+Cx^5$