The Laplace transform takes the di erential equation for a function y and forms an associated algebraic equation to be solved for L(y). Then, one has to take the inverse Laplace transform to get y.

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By the Fundamental Theorem of Calculus, l′ x 1/x = Given y 0 the function φ x l xy therefore satis es > = 1 1 φ′ x y l′ xy y and φ 1 = = xy = x Hence

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V into W is a function L: V ! W such that L(x+y) = L(x)+L(y) and L(cx) = cL(x) fo all reals c and vectors x; y. Equivalently, L(ax + by) = aL(x) + bL(y), for all rea

Application to differential equations (2) Expression for Laplace transform: Set Y ≡ L(y). Then: s2 − s − 2 Y (s) + (1 − s)y(0) − y0(0) = 0