If `y=x^2 e^x`, show that `y_n=1/2n(n-1)y_2-n(n-2)y_1+1/2(n-1)(n-2)y`

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

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

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

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 x + y = 1, show that D ^ (n) (x ^ (n) y ^ (n)) = n! [Y ^ (n) - (nC_ (1)) ^ (2) y ^ (n -1) x + (nC_ (2)) ^ (2) y ^ (n-2) x ^ (2) + ...... + (- 1) ^ (n) x ^ (n)]

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 S_(n)=(x+y)+(x^(2)+xy+y^(2))+(x^(3)+x^(2)y+y^(2)x+y^(3))+…n terms then prove that (x-y)S_(n)=[(x^(2)(x^(n)-1))/(x-1)-(y^(2)y^(n)-1)/(y-1)] .

If y=a x^(n+1)+b x^(-n) , then x^2(d^2y)/(dx^2)= n(n-1)y (b) n(n+1)y (c) n y (d) n^2y

If y=a x^(n+1)+b x^(-n),t h e nx^2(d^2y)/(dx^2) is equal to n(n-1)y (b) n(n+1)y (c) n y (d) n^2y