I have the solutions but it doesn't have detailed explantion. take as much time as you need. Thanks in advance!

2. Find f(t) = C'{F(s)} given that 2 2s (i) F(s) = (ii) F(s) = e-3s 352 + 27 $2 + 35 - 183. Use Laplace transforms to solve the initial value problem y" + 6y'+ 13y = S(t- 4), y(0) = 0, y'(0) = 0. ow 5. Use the chain rule to find when u =1, v =-2, if w = y sinz+ xIny and x =uv, au 4u y = (2 , Z = V-U.The temperature at a point (x,y,z)on a surface is given by T(x,y,z) : 3y2 4yz+5xyz. (i) Find the rate of change of the temperature at the point P(0,1,71) in the direction towards the point Q(6,2,3). (ii) In what direction does the temperature T increase most rapidly at the point P? (iii) What is the maximum rate of change of the temperature Tat the point P