Solve the following initial value problem: `(d r)/(dt)=\ -r t ,\ r(0) = r_0`

Solve the following initial value problem: (dr)/(dt)=-rt,y(2)=0

Solve the following initial value problems: y'=secy, y(0)=0

Solve each of the following initial value problem: (1+y^(2))dx+(x-e^(-tan)hat r((-1)))dy=0,y(0)=0

Show that the function phi , defined by phi(x)= cos x(x in R) , satisfies the initial value problem (d^(2)y)/(dx^2)+y=0, y(0)=1, y'(0)=0 .

Verify that each of the following functions y:R rarr R , as defined below, is the solution of the accompanying initial value problems : y= cos x (x in R): y''+y=0, y(0)=1, y'(0)=0 .

Verify that each of the following functions y:R rarr R , as defined below, is the solution of the accompanying initial value problems : y=e^(x):y'=y, y(0)=1

Solve the foregoing problem for the case of two concentric spheres of radii R_1 and R_2 and temperatures T_1 and T_2 .

Prove the following: sum_(r =0)^(n)(-1)^(r ) (3r+5).C_(r )=0

Write down the image of the following points w.r.t. To the axis or the plane as indicated : (d) (5, 0, -1) w.r.t. XY -plane

Differentiate the following w.r.t. x: sinx^0