Program Explanation Given information: The differential equation is d x d t = 1 − x 2 and the initial condition is x ( 0 ) = 3 . Calculation: The given differential equation is d x d t = 1 − x 2 . First separate the variables of the differential equation d x d t = 1 − x 2 as, d x d t = 1 − x 2 d x 1 − x 2 = d t d x ( 1 + x ) ( 1 − x ) = d t Use the partial fraction method to solve the given differential equation. Consider the coefficient of d x from above equation 1 ( 1 + x ) ( 1 − x ) 1 ( 1 + x ) ( 1 − x ) = a 1 + x + b ( 1 − x ) .....(1) Now solve the above equation as, 1 ( 1 + x ) ( 1 − x ) = a 1 + x + b ( 1 − x ) 1 x ( 10 − x ) = a ( 1 − x ) + b ( 1 + x ) ( 1 + x ) ( 1 − x ) 1 = a − a x + b + b x 1 = a + b + x ( − a + b ) Compare the coefficient of x and constant to obtain the value of a and b . a + b = 1 , b − a = 0 Therefore, the value of a and b is, a = 1 2 , b = 1 2 Now substitute 1 2 for a and 1 2 for b in equation (1). 1 ( 1 + x ) ( 1 − x ) = 1 2 ( 1 + x ) + 1 2 ( 1 − x ) Now integrate the differential equation on both sides. ∫ d x 1 − x 2 = ∫ d t ∫ ( 1 2 ( 1 − x ) + 1 2 ( 1 + x ) ) d x = ∫ d t − 1 2 ln ( 1 − x ) + 1 2 ln ( 1 + x ) = t + c 1 2 ln 1 + x 1 − x = t + c Now simplify the further equation. 1 + x 1 − x = c e 2 t 1 + x = c e 2 t ( 1 − x ) 1 + x = c e 2 t − x c e 2 t 1 + x + x c e 2 t = c e 2 t Further simplify the above equation

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Chapter 2.1, Problem 3P

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Program Description: Purpose of the problem is to find the particular solution of the differential equation dxdt=1−x2 and then graph the solution.

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Chapter 2.1, Problem 2P

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Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

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Program Description: Purpose of the problem is to find the particular solution of the differential equation d x d t = 1 − x 2 and then graph the solution.

Solution Summary: The author explains that the problem is to find the particular solution of the differential equation and then graph the solution.

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Program Description: Purpose of the problem is to find the particular solution of the differential equation dxdt=1−x2 and then graph the solution.

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Chapter 2 Solutions