Explanation Given: y ( 0 ) = 0 Approach: Simpson’s Rule: The definite integral ∫ f ( x ) d x in the interval [ a , b ] can be approximated as, ∫ a b f ( x ) d x = p 3 ( y 0 + 4 y 1 + 2 y 2 + 4 y 3 + … 2 y 2 n − 2 + 4 y 2 n − 1 + y 2 n ) Here, n is the number of sub intervals. And, h = b − a n Procedure: The standard form of the equation is, d y d x + P ( x ) y = Q ( x ) Calculate the integrating factor. μ ( x ) = exp [ ∫ P ( x ) d x ] Multiply the standard form by μ ( x ) Integrate the obtained equation and solve for y by diving by μ ( x ) . Calculation: Consider the following equation. d y d x + sin 2 x 2 ( 1 + sin 2 x ) y = 1... ( 1 ) Compare equation ( 1 ) with the standard of differential equation. d y d x + P ( x ) y = Q ( x ) So, P ( x ) = sin 2 x 2 ( 1 + sin 2 x ) Integrating the above equation, ∫ P ( x ) = ∫ sin 2 x 2 ( 1 + sin 2 x ) d x = ∫ 2 cos x sin x 2 ( 1 + sin 2 x ) d x = ∫ cos x sin x ( 1 + sin 2 x ) d x ... ( 2 ) Let 1 + sin 2 x = t So, d t = 2 sin x cos x d x d x = d t 2 sin x cos x Substitute t for 1 + sin 2 x and d t 2 sin x cos x for d x in equation ( 2 ) ∫ P ( x ) = ∫ cos x sin x t ( d t 2 sin x cos x ) = 1 2 ∫ d t t = 1 2 ln ( t ) = ln ( t ) Substitute 1 + sin 2 x for t in the expression. ∫ P ( x ) = ln ( 1 + sin 2 x ) Calculate the integrating factor. μ ( x ) = e ∫ P ( x ) d x = e ln ( 1 + sin 2 x ) = 1 + sin 2 x Multiply by 1 + sin 2 x in equation ( 1 )

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Chapter 2.3, Problem 26E

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The solution of the differential equation dydx+sin2x2(1+sin2x)y=1 at x=1 using numerical integration.

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The solution of the differential equation d y d x + sin 2 x 2 ( 1 + sin 2 x ) y = 1 at x = 1 using numerical integration .

The solution of the differential equation dydx+sin2x2(1+sin2x)y=1 at x=1 using numerical integration.

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