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FormulaF - Math 421:01 and 02 Spring 2008 Final Exam Review...

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Math 421:01 and 02 — Spring 2008 Final Exam Review and Formula Sheet Trigonometry The formula e it D cos t C i sin t (with i 2 D 1 ) connects exponential and trigonometric functions. Consequences include the basic identities cos C ˇ/ D cos ˛ cos ˇ sin ˛ sin ˇ sin C ˇ/ D sin ˛ cos ˇ C cos ˛ sin ˇ that are used to prove the orthogonality of the functions f 1; . cos nx; sin nx W n D 1; 2; : : :/ g on the interval < x < . Laplace transforms The Laplace transform of a function of t is a function of a new variable s defined by ˇ f f .t/ g D Z 1 0 f .t/e st dt This definition shows that the Laplace transform is a linear operator . In describing properties of the Laplace transform, it is conventional to write F.s/ for ˇ f f .t/ g . Some useful properties (in addition to linearity) are f .t/ F.s/ 1 1 s t n 1 s n C 1 cos t s s 2 C 1 sin t 1 s 2 C 1 f 0 .t/ sF.s/ f .0/ f .bt/ 1 b F .s=b/ e at f .t/ F.s a/ tf .t/ F 0 .s/ f .t a/ U .t a/ e as F.s/ ı.t a/ e as .f g/.t/ F.s/G.s/ In the last formula, .f g/.t/ denotes the convolution of f .t/ and g.t/ which is defined by .f g/.t/ D Z t 0 f . /g.t / d
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Math 421:01 and 02, May 13-14, 2008, p. 2 Calculation of Laplace transforms or inverse transforms typically requires several steps taken from this table. To apply these rules, functions must be found so that the given expression has the form shown in a line of this table. Ordinary Differential Equations There are two families of differential equations that are easy to solve exactly . ) Linear equations with constant coefficients . Usually, the solutions have the form y D e cx . . ) Euler equations , whose terms are combinations with constant coefficients of x k d k y=dx k for several integer k . Usually, the solutions have the form y D x c . To solve these equations, try the correct form of solution. The equation reduces to an algebraic equation in c . If a root c is complex, the exponential is usually converted to a form containing trigonometric functions. For Euler equations, this also requires the identity x c D e c ln x . If the algebraic equation has repeated roots , this method does not give enough solutions. If c is a repeated root, a second solution is xe cx in the constant coefficient case, or x c ln x in the Euler equation case. A full characterization of solutions will not be given here, since it will be easy to check any tentative solution constructed from this principle. Fourier series The Fourier series of a function f .x/ on the interval p < x < p is given by a 0 2 C 1 X n D 1 a n cos n x p C b n sin n x p where a 0 D 1 p Z p p f .x/ dx a n D 1 p Z p p f .x/ cos n x p dx b n D 1 p Z p p f .x/ sin n x p dx . / The constant term a 0 =2 is the average value of f .x/ , and Z p p cos 2 n x p dx D p Z p p sin 2 n x p dx D p so that the Fourier series of f .x/ D 1 , f .x/ D cos n x=p or f .x/ D sin n x=p is a single term identical to f .x/ .
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