HP 40gs hp 40gs_user's guide_English_E_HDPMSG40E07A.pdf - Page 248
The following relations hold
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hp40g+.book Page 66 Friday, December 9, 2005 1:03 AM 14-66 ILAP is the inverse Laplace transform of a given expression. Again, the expression is the value of a function of the variable stored in VX. Laplace transform (LAP) and inverse Laplace transform (ILAP) are useful in solving linear differential equations with constant coefficients, for example: y″ + p ⋅ y′ + q ⋅ y = f(x) y(0) = a y′(0) = b The following relations hold: ∫ LAP(y)(x) = +∞ e-x ⋅ ty(t)dt 0 ∫ ILAP(f)(x) = 2---1-i--π-- ⋅ ezxf(z)dz c where c is a closed contour enclosing the poles of f. The following property is used: LAP(y′)(x) = - y(0) + x ⋅ LAP(y)(x) The solution, y, of: y″ + p ⋅ y′ + q ⋅ y = f(x), y(0) = a, y′(0) = b is then: I L A P ⎝⎛ L-----A----P----(--f--(---xx---)2--)--+--+---p--(-x-x----++-----qp a-----+----b--⎠⎞ Example To solve: y″-6 ⋅ y′ + 9 ⋅ y = x ⋅ e3x, y(0) = a, y′(0) = b c type: LAP(X · EXP(3 · X)) The result is 1 x2 - 6x + 9 Computer Algebra System (CAS)