Initial Value Problems for Ordinary Differential Equations

Heinz Rutishauser

Chapter Chapter 8 in Lectures on Numerical Mathematics, 1990, pp 208-277 from Springer

Abstract: Abstract It is a well-known fact that differential equations occurring in science and engineering can generally not be solved exactly, that is, by means of analytical methods. Even when this is possible, it may not necessarily be useful. For example, the second-order differential equation with two initial conditions, (1) $$y'' + 5y' + 4y = 1 - {e^x},\,\,\,\,\,\,\,\,\,\,\,y\left( 0 \right) = y'\left( 0 \right) = 0,$$ has the exact solution (2) $$y = \frac{1}{4} - \frac{1}{3}x{e^{ - x}} - \frac{2}{9}{e^{ - x}} - \frac{1}{{36}}{e^{ - 4x}},$$ but when this formula is evaluated, say at the point x =.01, one obtains with 8-digit computation $$y = .25 - 00330017 - .22001107 - .02668860 = .00000016,$$ which is no longer very accurate.

Keywords: Ordinary Differential Equation; Trapezoidal Rule; Amplification Factor; Difference Formula; Linear Multistep Method (search for similar items in EconPapers)
Date: 1990
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DOI: 10.1007/978-1-4612-3468-5_8

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