ODE 602S
Ordinary Differential Equations
January 2024
l. (a) i. Solve the following initial value problem:
y'()X = y(x)+x y(2) = 8, x>0
X'
(3)
ii. Hence or otherwise find y(x) at x = 8
(2)
(b) Solve the following initial value problems:
y'(x) + 20 xy(x) + 1 = 0, y(0) = 20, x 2: O
(5)
(c) 1. Suppose a returning student brings the flu virus to his/her boarding house
college campus of 5,000 students. Suppose further that the rate at which the
virus spreads is proportional not only to the number of infected students but
also to the number of students not infected. Determine the number of infected
students after 7 days if it is observed that after 5 days the number of infected
students is 70.
(5)
11. A tank initially holds 300 liters of liquid, with 20 grams of dissolved salt. Brine,
with a salt concentration of 1 gram per liter, is continuously pumped into the
tank at a rate of 4 liters per minute. Simultaneously, a well-mixed solution
is pumped out of the tank at the same rate. Determine the function N(t),
representing the number of grams of salt in the tank at time t.
(5)
2. (a) If y1 and y2 are two solutions of second order homogeneous differential equation of
the form
y"(x) + p(x)y'(x) + q(x)y(x) = J(x)
where p(x) and q(x) are continuous on an open interval I, derive the formula for
u(x) and v(x) by using variation of parameters.
(6)
(b) If
Y1(x) = 2x + 1, W(Y1, Y2) = 2x2 + 2x + 1, Y2(0) = 0
find Y2(x)
(7)
(c) Solve
8x2y"(x) + 16xy'(x) + 2y(x) = 0
(7)
3. Solve the following using Laplace Transform
(a)
2
d y(x)
+ 2dy(t)
+ 5y(t)
=
e-t sin t
dt2
dt
'
y(0) = 0,
y'(0) = 1
(b)
dxd~t)- y(t) = et, dy(t) + X (t) = Sl.n t, X (0) = 1, y (O) = 0
(7)
2
(7)