Q 1. An object rotates with angular acceleration, a defined as a = —r- where cu is the angular velocity Given that when t = 0, uj = u;0 show that uj = cu0 + ott Also the angular
at
velocity u; is defined as u; = — where 0 is the angular displacement. Given that when t = 0,0 = 0, then show that 0 = u)0t + — at2. [2marks]
at 2
1
Q 2. The pressure P of the atmosphere at an altitude 2 is given by = -KV 1 where k ± 0 where 7 is the specific heat constant and k is a constant. Show that
dz
1
7
z =
IV
7
7-1
+
K
[ 1 mark]
di
Q 3. By applying Kirchhoff’s voltage law to a series RL circuit we obtain the differential equation 1 x 10 6— 4- (3 x 106)i = 10ef where i = i(t) is the current through the
circuit. Given the initial condition i(0) = 0 determine an expression for the current i. [2marks]
dy
Q 4. If tail x sin x is an integrating factor of the differential equation — h Py = Q then P =? [1 mark]
dx
Q 5. Determine the constant a of the differential equation such that the differential equation is exact and hence solve it (x2 4- 3xy)dx + (ax2 + 4y)dy = 0 [ lmark]
Q 6. Find the orthogonal trajectories of the function^/ = x + ce~x, which is passing through (—2,0) [ 2 marks]
Q 7. Solve the non linear differential equation ey(-^ + l) = er [1 mark]