If `y=sin^(-1)x` ,then prove that `(1-x^(2))y_(n+2)-(2n+1)xy_(n+1)-n^(2)y_(n)=0`

If y=(sin^(-1)x)^2+(cos^(-1)x)^2 , then prove that (1-x^2)y_2-xy_(1)-4=0.

If I_n=int_0^ooe^(-x)x^(n-1)log_exdx , then prove that I_(n+2)-(2n+1)I_(n+1)+n^2I_n=0

If x^m y^n=(x+y)^(m+n) , prove that (d^2y)/(dx^2)=0

If y^(2)=ax^(2)+2bx+c and u_(n)= int (x^(n))/(y)dx , prove that (n+1)a u_(n+1)+(2n+1)bu_(n)+(n)c u_(n-1)=x^(n)y and deduce that au_(1)=y-b u_(0), 2a^(2)u_(2)=y(ax-3b)-(ac-3b^(2))u_(0) .

If x cos A + y sin A = m and x sin A - y cos A = n, then prove that : x^(2) + y^(2) = m^(2) + n^(2)

If x^m y^n=(x+y)^(m+n) , prove that (d^2y)/(dx^2)=0 .

If x^my^n = (x + y)^(m+n) , prove that (d^2y)/(dx^2)=0 .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) - C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n)," prove that " 1^(2)*C_(1) + 2^(2) *C_(2) + 3^(2) *C_(3) + …+ n^(2) *C_(n) = n(n+1)* 2^(n-2) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that (1*2) C_(2) + (2*3) C_(3) + …+ {(n-1)*n} C_(n) = n(n-1) 2^(n-2) .