If `0ltxlt1` prove that `y=xlnx-x^2/2+1/2` is a function such that `(d^2y)/dx^2gt0`

If y=sin^-1x ,then show that (1-x^2)(d^2y)/dx^2-xdy/dx=0

For each of the following initial value problems verify that the accompanying functions is a solution. (i) x(dy)/(dx)=1, y(1)=0 => y=logx (ii) (dy)/(dx)=y , y(0)=1 => y=e^x (iii) (d^2y)/(dx^2)+y=0, y(0)=0, y^(prime)(0)=1 => y=sinx (iv) (d^2y)/(dx^2)-(dy)/(dx)=0, y(0)=2, y^(prime)(0)=1 => y=e^x+1 (v) (dy)/(dx)+y=2, y(0)=3 => y=e^(-x)+2

If y=tan^-1x , prove that , (x^2+1)(d^2y)/(dx^2)+2xdy/dx=0 .

If (x-a)^(2)+(y-b)^(2)=c^(2) , for some c gt 0 , prove that ([1+((dy)/(dx))^(2)]^(3/2))/((d^(2)y)/(dx^(2))) is a constant independent of a and b.

If (x-a)^(2)+(y-b)^(2)=c^(2) , for some c gt 0 , prove that ([1+((dy)/(dx))^(2)]^(3/2))/((d^(2)y)/(dx^(2))) is a constant independent of a and b.

If (x-a)^(2)+(y-b)^(2)=c^(2) , for some c gt 0 , prove that ([1+((dy)/(dx))^(2)]^(3/2))/((d^(2)y)/(dx^(2))) is a constant independent of a and b.

If (x-a)^(2)+(y-b)^(2)=c^(2) , for some c gt 0 , prove that ([1+((dy)/(dx))^(2)]^(3/2))/((d^(2)y)/(dx^(2))) is a constant independent of a and b.

If x=log theta ,theta gt 0and y= (1)/( theta ),then (d^(2) y)/(dx^(2))=

If y=sin(logx) prove that x^2(d^2y)/dx^2+xdy/dx+y=0 .

If y=sin^(-1) x, prove that (1-x^2)(d^2y)/(dx^2)-x(dy)/(dx)=0