Mathematical Modeling with Multidisciplinary Applications

PROBLEM SOLUTIONS

SOLUTIONS FOR CHAPTER 1

1.1 Let y be the axis perpendicular to the path of its movement towards the east. From Newton’s second law, we know that

where ν is the vertical velocity, and ω is the angular velocity of the Earth.

   Here we have assume that the horizontal velocity due to the Coriolis effect is small, and no air is present. From basic physics, we know that ν = gt where g is the acceleration due to gravity. We have .

   Now the governing equation becomes

By integrating ÿ once, we have

Integrating it again, we obtain

Since ω = 2π/(24 × 3600) radian/s, g = 9.8 m/s², we have y ≈ 7.7 mm for h = 50 m.

1.2 The solution is y(x) = A/x + Bx² where A and B are arbitrary constants.

1.3 Using a trial solution u ~ exp(rt), we have

or

Its solutions are

Clearly, Δ = η² − 1 ...