If the solution of the equation `(d^(2)x)/(dt^(2))+4(dx)/(dt)+3x = 0` given that for `t = 0, x = 0 and (dx)/(dt) = 12` is in the form `x = Ae^(-3t) + Be^(-t)`, then

The solution of differential equation (dt)/(dx) = (t[d/dx{g(x)}]-t^(2))/(g(x) ) is

Let y=x^(3)-8x+7 and x=f(t)." If "(dy)/(dt)=2 and x=3 at t = 0," then "(dx)/(dt) at t = 0 is given by

Number of solution of the equation (d)/(dx)int_(cos x)^( sin x)(dt)/(1-t^(2))=2sqrt(2)in[0,pi] is

Let x=x(t) and y=y(t) be solutions of the differential equations (dx)/(dt)+ax=0 and (dy)/(dt)+by=0] , respectively, a,b in R .Given that x(0)=2,y(0)=1 and 3y(1)=2x(1) ,the value of "t" ,for which, [x(t)=y(t) ,is :

(d)/(dx) [int_(0)^(x^(2))(dt)/(t^(2) + 4)] = ?

The number of integral solutions of the equation 4int _(0)^( infty)(In" t"dt)/(x^(2)+t^(2))-piIn 2 =0, x gt 0, is

The number of integral solutions of the equation 4int _(0)^( infty)(In" t"dt)/(x^(2)+t^(2))-piIn 2 =0, x gt 0, is

If y = int_(0)^(x) (t^(2))/(sqrt(t^(2)+1))dt then (dy)/(dx) at x=1 is