If `y=e^(asin^(-1)x)` then show that `(1-x^2)y_(n+2)-(2n+1)xy_(n+1)-(n^2+a^2)y_n=0`

If 2x=y^(1/m)+y^(-1/m) , then prove that (x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0 .

If y=ax^(n+1)+bx^(-n) then show that x^(2)y''=n(n+1)y .

If y=px^(n)+qx^(-n) , show that x^(2)y_(2)+xy_(1)-n^(2)y=0 .

If y = (x+ sqrt(1+x^2))^n then prove that (1+x^2)y_2+xy_1 = n^2y .

If y = sin(m sin^(-1)x) , then (1-x^(2))y_(2) - xy_(1) is equal to (Here, y_(n) denotes (d^(n)y)/(dx^(n)) )

If y=x^(2)e^(x) ,show that y_(n)=(1)/(2)n(n-1)y_(2)-n(n-2)y_(1)+(1)/(2)(n-1)(n-2)}

If y=x^(n-1)ln x then x^(2)y_(2)+(3-2n)xy_(1) is equal to -(n-1)^(2)y(b)(n-1)^(2)y-n^(2)y(d)n^(2)y

If y=x^(n-1) log x , prove that (x^2y_2)+(3-2n)xy_1+(n-1)^2 .y=0 where y_1=dy/dx and y_2=(d^2)/(dx^2) .