Find a fundamental set of solutions {y_(1),y_(2)}of the equation x^(2) (d^(2)y)/(dx^(2)) - 13x (dy)/(dx) + 49y=0 in the interval x>0

Question

Find a fundamental set of solutions {
`y_(1),y_(2)`}of the equation
`x^(2) (d^(2)y)/(dx^(2)) - 13x (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.