Page 104 - MATLAB an introduction with applications

MATLAB Basics ——— 89

2
(b) y′ = 5x sin (y) with initial condition y(0) = π/5.
2
(c) y′ = 7x cos (y) with initial condition y(0) = 2.
(d) y′ = –5x + y with initial condition y(0) = 3.
–5x
(e) y′ = 3y + e with initial condition y(0) = 2.
P1.27: For the following differential equations, use MATLAB to find x(t) when (a) all the initial conditions
are zero, (b) x(t) when x (0) = 1 and x (0)=–1.
2
2
dx dx dx dx
(a) +10 + 5x = 11 (b) –7 3x = 5
dt 2 dt dt 2 dt
2
2
dx dx dx dx
(c) +3 + 7x = –15 (d) + + 7x = 26
dt 2 dt dt 2 dt
P1.28: Figure P1.28 shows a water tank (shaped as an inverted frustum cone with a circular hole at the
bottom on the side).
R=0.5m

3m

y

r =0.025m
h
R=2 m

Fig. P1.28 Water tank

The velocity of water discharged through the hole is given by v = 2gy where h = height of the water and
2
g = acceleration due to gravity (9.81 m/s ). The rate of discharge of water in the tank as the water drains out
dy 2gyr 2
through the hole is given by: h where y = height of water and r = radius of the hole. Write a

dt (2 0.5 ) 2 h
y
MATLAB program to solve and plot the differential equation. Assume, that the initial height of the water is 2.5 m.
P1.29: An airplane uses a parachute (see Fig. P1.29) and other means of braking as it slow down on the
2
runway after landing. The acceleration of the airplane is given by a = – 0.005 v – 4 m/s 2

v
x

Fig. P1.29

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