Explanation Procedure used: Solution to a differential equation: The n th-order ordinary differential equation is one dependent variable by the general form, F ( x , y , y ' , ... , y ( n ) ) = 0 Where F is a real valued function of n + 2 variables: x , y , y ' , ... , y ( n ) . The solution of an n th-order ordinary differential equation is a function ϕ that possesses at least n derivatives and satisfying the differential equation on interval I , i.e. F ( x , ϕ ( x ) , ϕ ' ( x ) , ... , ϕ ( n ) ( x ) ) = 0 for all x in I . The Fundamental Theorem of Calculus states that the function F defined by, F = ∫ a x g ( t ) d t on the interval [ a , b ] , a ≤ x ≤ b , is differentiable on ( a , b ) and, F ′ ( x ) = d d x ∫ a x g ( t ) d t = g ( x ) Calculation: Consider the given differential equation d y d x + 2 x y = 1 and the given function y = e − x 2 + e − x 2 ∫ 0 x e t 2 d t

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 1.1, Problem 28E

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To verify: The function y=e−x2+e−x2∫0xet2dt is a solution to the differential equation dydx+2xy=1.

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Chapter 1.1, Problem 27E

Chapter 1.1, Problem 29E

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To verify: The function y = e − x 2 + e − x 2 ∫ 0 x e t 2 d t is a solution to the differential equation d y d x + 2 x y = 1 .

Solution Summary: The author explains the fundamental theorem of calculus, which states that the function F is differentiable on the interval (a,b).

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To verify: The function y=e−x2+e−x2∫0xet2dt is a solution to the differential equation dydx+2xy=1.

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Chapter 1 Solutions