`(d y)/(dx)+x/(1+x^2)\ y=1/(2x(1+x^2))`

If y=sin^(-1)x, show that (d^(2)y)/(dx^(2))=(1)/((1-x^(2))^(3/2))

Solve :(d^(2)y)/(dx^(2))=(2x)/((1+x^(2))^(2)); given that x=0,(dy)/(dx)=1 and y=2

For each of the following differential equations verify that the accompanying functions a solution. Differential Function (a) x(dy)/(dx)=y, y=a x (b) x+y(dy)/(dx)=0, y=+-sqrt(a^2-x^2) (c) x(dy)/(dx)y=y^2, y=a/(x+a) (d) x^3(d^2y)/(dx^2)=1, y=a x+b+1/(2x) (e) y=((dy)/(dx))^2, y=1/4(x+-a)^2

If y = e^(a cos^(-1)x), -1 le x le 1 show that (1 - x)^(2) (d^2 y)/(dx^2) - x (dy)/(dx) - a^(2)y = 0 .

If y=sin^(-1)x , show that (d^2y)/(dx^2)=x/((1-x^2)^(3//2))

If e^(y)(x+1)=1, show that (d^(2)y)/(dx^(2))=((dy)/(dx))^(2) If y=sin(2sin^(-1)x), show that ((1-x^(2))d^(2y))/(dx^(2))=x(dy)/(dx)-4y

If y = e^(2tan^-1 x) , then show that (1+x^2)^2 (d^2y)/(dx^2) + 2x(1+x^2)dy/dx = 4y

If y = e^(4tan^-1 x) , then show that (1+x^2)^2 (d^2y)/(dx^2) + 2x(1+x^2)dy/dx = 16y