Solve the boundary value problem x^(2) (d^(2)y)/(dx^(2)) - 15x (dy)/(dx) + 64y=0 where y(1)=0 and y(2)=1

Question

Solve the boundary value problem


`x^(2) (d^(2)y)/(dx^(2)) - 15x (dy)/(dx) + 64y=0`
where y(1)=0 and y(2)=1

Answer 1

Another Cauchy-Euler ODE. Take
`y=x^r` and you get


`r(r-1)x^r-16rx^r+64x^r=0`

`r^2-17r+64=0`

`\implies r=\frac{17}2\pm\frac{√(33)}2`

so that the general solution is


`y=C_1x^((17+√(33))/2)+C_2x^((17-√(33))/2)`

With the given boundary conditions, you have


`y(1)=0\implies 0=C_1+C_2`

`y(2)=1\implies 1=C_12^((17+√(33))/2)+C_22^((17-√(33))/2)`

`\implies C_1=(2^((-17+√(33))/2))/(2^(√(33))-1),C_2=-(2^((-17+√(33))/2))/(2^(√(33))-1)`