If `e^(y)(x + 1) = 1` show that `(d^2 y)/(dx^2) = ((dy)/(dx))^(2)`.

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

If y=e^(3logx) , then show that (dy)/(dx)=3x^2

Find the order and degree, if defined, of each of the following differential equations: (i) (dy)/(dx) - cos x = 0 (ii) xy (d^(2)y)/(dx^(2)) + x((dy)/(dx))^(2) - y (dy)/(dx) = 0 (iii) y''' + y^(2) + e^(y)' = 0

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 .

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

If Y=Ae^(mx) + Be^(nx) show that: (d^(2)y)/(dx^(2)) -(m+n) (dy)/(dx) + mny=0

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

If y = 3e^(2x) + 2e^(3x) , prove that (d^2y)/(dx^2) - 5(dy)/(dx) + 6y = 0