Q-7quad " If "y=x^(2)e^(x)," prove that "y_(n)=(1)/(2)n(n-1)y_(2)-n(n-2)y_(1)+(1)/(2)(n-1)(n-2)y

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=ax^(n+1)+bx^(-n), then x^(2)(d^(2)y)/(dx^(2))=n(n-1)y(b)n(n+1)y(c)ny(d)n^(2)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 .

(x^(2^(n-1))+y^(2^(n-1)))(x^(2^(n-1))-y^(2^(n-1)))=

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

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=((1+x)/(1-x))^(n) , prove that (1-x^(2))(d^(2)y)/(dx^(2))=2(n+x)(dy)/(dx).

If sqrt(1-x^(2n))+sqrt(1-y^(2n))=a^(n)(x^(n)-y^(n)) , prove that, (dy)/(dx)=((x)/(y))^(n-1).sqrt((1-y^(2n))/(1-x^(2n))

(x^(2^n)-y^(2^n))/(x^(2^(n-1))+y^(2^(n-1)))=