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

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ISBN: 8220100802454

Author: Calvis

Publisher: PEARSON

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Linear Programming Concepts

Linear programming is a famous mathematical modeling tool for determining the best distribution of scarce resources among competing demands. Machinery, time, personnel, raw materials, money, and space are all resources. It is used to find the most optima…

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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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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.

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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