To determine The value of x as a function of t using the given initial value problem. Explanation Given information: The initial value problem is ( t 2 + 2 t ) d x d t = 2 x + 2 . The given initial value condition is x ( 1 ) = 1 . The value of t and x are greater than 0. Formula: ∫ 1 x d x = ln | x | + C Calculation: Write the initial value problem as below. ( t 2 + 2 t ) d x d t = 2 x + 2 Simplify it as below. d x ( 2 x + 2 ) = d t ( t 2 + 2 t ) Apply integration operator as below. ∫ d x ( x + 1 ) = ∫ 2 d t ( t 2 + 2 t ) ln | x + 1 | = ∫ 2 d t t ( t + 2 ) Express the integrand as a sum of partial fractions with undetermined coefficients. 2 t ( t + 2 ) = A t + B ( t + 2 ) (1) Simplify the above equation. 2 t ( t + 2 ) = A ( t + 2 ) + B t t ( t + 2 ) 2 = A ( t + 2 ) + B t 2 = t ( A + B ) + 2 A (2) Equate the constant term on both the sides of equation (2). 2 A = 2 A = 1 Equate the coefficient of the t term on both the sides of equation (2)

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Chapter 8.5, Problem 69E

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The value of x as a function of t using the given initial value problem.

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Chapter 8.5, Problem 68E

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