Explanation Given: Lunar Orbit- The motion of a moon moving in a planar orbit about a planet is governed by the equations d 2 x d t 2 = − G m x r 3 d 2 y d t 2 = − G m y r 3 here r : = ( x 2 + y 2 ) 1 / 2 , G is the gravitational constant, and m is the mass of the planet. When x ( 0 ) = 1 x ′ ( 0 ) = y ( 0 ) = 0 y ′ ( 0 ) = 1 the motion is a circular orbit of radius r 1 and period t 2 π . Approach: Convert the given initial value problem into initial value problem for a system in normal form. Set, x 1 ( t ) = x ( t ) x 2 ( t ) = x ′ ( t ) x 3 ( t ) = y ( t ) x 4 ( t ) = y ′ ( t ) ( 1 ) Assume G m = 1 . Calculation: Differentiate system of equations ( 1 ) with respect to t . x ′ 1 ( t ) = x 2 ( t ) x ′ 2 ( t ) = x ′′ ( t ) x ′ 3 ( t ) = x 4 ( t ) x ′ 4 ( t ) = y ′′ ( t ) ( 2 ) Do substitution from equation ( 2 ) in the given initial value problem. x ′′ ( t ) = − G m x r 3 y ′′ ( t ) = − G m y r 3 x ( 0 ) = 1 x ′ ( 0 ) = y ( 0 ) = 0 y ′ ( 0 ) = 1 Here, r = ( x 2 + y 2 ) 1 / 2 G m = 1 After substitution x ′ 2 ( t ) = − G m r 3 x 1 ( t ) = − x 1 ( t ) ( x 1 2 ( t ) + x 3 2 ( t ) ) 3 / 2 x ′ 4 ( t ) = − G m r 3 x 3 ( t ) = − x 3 ( t ) ( x 1 2 ( t ) + x 3 2 ( t ) ) 3 / 2 To determine (b) To compute: One orbit of the moon and check whether the approximations agree with the fact that the orbit is a circle of radius 1 .

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 5.3, Problem 28E

To determine

(a)

To express:

The governing equations as a first-order system in normal form.

To determine

(b)

To compute:

One orbit of the moon and check whether the approximations agree with the fact that the orbit is a circle of radius 1.

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Chapter 5.3, Problem 27E

Chapter 5.3, Problem 29E

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Fundamentals of Differential Equations and Boundary Value Problems

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(a) To express: The governing equations as a first-order system in normal form.

Solution Summary: The author explains the governing equations as a first-order system in normal form.