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Solutions of ODEs almost always involve definite or indefini

Solutions of ODEs almost always involve definite or indefinite integrals, and it is nice when we… Show more Solutions of ODEs almost always involve definite or indefinite integrals, and it is nice when we can evaluate those integrals in terms of other elementary functions. But much more often, we cannot do this, and these integrals remain “unevaluated” in our solutions. (Actually, this is not such a bad thing, since it is at least possible to estimate values of definite integrals using, for example, Riemann sums.) Here are some examples of definite integrals that cannot be evaluated in terms of elementary functions and that arise frequently in physics and engineering. 1. The error function is defined by erf(x)=2/pi????x0e?t2dt (6) and it frequently arises in engineering applications and statistics. Show that erf(x) solves the IVP dydx=2/pi???e?x2 y(0)=0 (7) Use Euler%u2019s method on IVP (7) with n = 4 steps starting at x = 0 to approximate erf(1). 2. The sine integral is defined by Si(x)=?x0(sin(t)/t)dt (8) and it frequently arises in signal processing. Show that Si(x) solves the IVP ddydx=sinx y(0)=0 (9) Use Euler’s method on IVP (9) with n = 4 steps starting at x = 0 to approximate Si(pi). (Note: Using Euler’s method this way to estimate the integrals (6, 8) is the same as using Riemann sums.) ? • Show less

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