Abstracts
For homogeneous linear first-order differential equations, it is shown that Picard’s method of successive approximations is effective to furnish a closed-form solution even if the coefficient is an arbitrary function.
Keywords:
Picard’s method; successive approximations; first-order differential equation
Para equações diferenciais lineares de primeira ordem homogêneas, é demonstrado que o método de sucessivas aproximações de Picard é eficaz para fornecer uma solução em forma fechada mesmo quando o coeficiente é uma função arbitrária.
Palavras-chave:
Método de Picard; sucessivas aproximações; equação diferencial de primeira ordem
In a recent didactic paper, Diniz [1[1] E.M. Diniz, Rev. Bras. Ens. Fis. 45, e20230054 (2023).] presented twelve different ways to solve the well-known simple harmonic oscillator problem. Among these methods lies Picard’s method of successive approximations, commonly utilized for resolving particular cases of first-order differential equations (see, e.g. [2[2] E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956)., 3[3] F.B. Hildebrand, Introduction to numerical analysis (Dover, New York, 1956)., 4[4] M. Tenenbaum and H. Pollard, Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences (Dover, New York, 1963)., 5[5] W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 2001)., 6[6] R.P. Agarwal and D. O’Regan, An Introduction to Ordinary Differential Equations (Springer, New York, 2008), 8 ed., 7[7] H.J. Ricardo, A Modern Introduction to Differential Equations (Academic Press, London, 2021), 3 ed.]). Here, after providing a succinct overview of this method, we demonstrate its effectiveness in yielding a closed-form solution for a homogeneous linear first-order differential equation, even when the coefficient is an arbitrary function.
Differential equations akin to
can be reformulated as integral equations:These integral equations can be solved iteratively. In Picard’s method, we derive a sequence of functions {yn (x)}n = 0,1,2,...,N, each satisfying the condition yn(x)|x = x0 = y0. It is supposed that there is an interval about y0 on which this sequence approaches the solution y(x) as N → ∞ and that it is the only continuous solution which does so (see, e.g. [2[2] E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956)., 3[3] F.B. Hildebrand, Introduction to numerical analysis (Dover, New York, 1956)., 4[4] M. Tenenbaum and H. Pollard, Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences (Dover, New York, 1963)., 5[5] W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 2001)., 6[6] R.P. Agarwal and D. O’Regan, An Introduction to Ordinary Differential Equations (Springer, New York, 2008), 8 ed., 7[7] H.J. Ricardo, A Modern Introduction to Differential Equations (Academic Press, London, 2021), 3 ed.]). Because each succeeding function improves the prior one, this method is termed the method of successive approximations. The initial approximation is y0 (x) = y0. Subsequent approximations are obtained asresulting inFor the general homogeneous linear first-order differential equation
where y(0) = y0 and the coefficient Q(x) is an arbitrary function, Picard’s method of successive approximations yieldswith yn(0) = y0. The first approximation is y0 (x) = y0, and the subsequent approximations follow suit: so thatwhere we have defined x0 = x. This can be compactly written aswhereThe definite integralis depicted in Figure 1 as an integral over the triangle above the dashed line x2 = x1 for 0 < x2 < x. The first integral is over the area of a horizontal slice of width dx2 ranging from 0 to x1 whereas the second integral adds up all the contributions from these horizontal slices from 0 to x1. Because x1 and x2 are dummy variables, I2(x) can also be written asNow, we can see an integral of the very same integrand over the triangle below the line x2 = x1. The first integral is over the area of a vertical slice of width dx1 ranging from 0 to x2. The second integral adds up all the vertical slices from 0 to x. Concisely, I2(x) represents half of the integral covering the square with 0 < x1 < x and 0 < x2 < x, i.e.It is instructive to note that, even without resorting to geometry, this result can be analytically derived by identifyingand subsequently applying integration by parts of u(x) du (x)/dx.Graphical representation of the double integral is over the triangle above the dashed line x2 = x1 for 0 < x2 < x, and is represented over the triangle below the dashed line for 0 < x1 < x. Both are equivalent to half of the integral over a square with sides equal to x.
In general,
because there are k! identical terms of the type (13), corresponding to the k! possible ways of interchanging the dummy variables x1, x2, ..., xk. As a result,such that equalswhich represents the general solution of (5).It can be confirmed that incorporating a constant nonhomogeneous term into equation (5) poses no additional challenge for Picard’s method. With diligent application of Leibniz’s theorem for differentiation under the integral sign and integration by parts repeatedly, we can smoothly include any arbitrary nonhomogeneous term.
References
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[1]E.M. Diniz, Rev. Bras. Ens. Fis. 45, e20230054 (2023).
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[2]E.L. Ince, Ordinary Differential Equations (Dover, New York, 1956).
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[3]F.B. Hildebrand, Introduction to numerical analysis (Dover, New York, 1956).
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[4]M. Tenenbaum and H. Pollard, Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences (Dover, New York, 1963).
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[5]W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 2001).
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[6]R.P. Agarwal and D. O’Regan, An Introduction to Ordinary Differential Equations (Springer, New York, 2008), 8 ed.
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[7]H.J. Ricardo, A Modern Introduction to Differential Equations (Academic Press, London, 2021), 3 ed.
Publication Dates
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Publication in this collection
15 July 2024 -
Date of issue
2024
History
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Received
31 Mar 2024 -
Accepted
10 May 2024