If `y=((a x+b))/((x^2+c))` , then show that `(2x y '+y)y'''=3y''(x y''+y'),` where `a , b , c` are constants and dashes denote differentiation w.r.t. `xdot`

If a cos (x + y) = b cos (x - y), show that (a + b)tan x = (a - b) cot y.

If y=(x^2)/2+1/2xsqrt(x^2+1)+(log)_esqrt(x+sqrt(x^2+1)) , prove that 2y=x y^(prime)+(log)_e y^(prime),w h e r ey ' denotes the derivative w.r.t xdot

If y= A cos 4x + Bsin 4x , A and B are constants then show that y_2 + 16y = 0

Show that y =a cos (log x)+b sin (logx),x le 0 is a soluton of the differential equation x^(2)y''+xy'+y=0

If y^(1/m)=(x+sqrt(1+x^2)),t h e n(1+x^2)y_2+x y_1 is (where y_r represents the r t h derivative of y w.r.t. xdot (a) m^2y (b) m y^2 (c) m^2y^2 (d) none of these

If cosx+cosy-cos(x+y)=3/2, then (a) x+y=0 (b) x=2y (c) x=y (d) 2x=y

Show that y=ax+(b)/(x), x ne 0 is a solution of the differential equation x^(2)y''+xy'-y=0.

If w(x,y,z) = x^2y+y^2z + z^2x,x,y,z in R, find the differential dw.

If y = A cos 4 x + B sin 4 x, A and B are constants then Show that y_2 + 16y =0