Laplace Transform of xsin(ax)+xcos(ax)

Laplace Transform of x sin ax + x cos ax shahbaz ahmed June 2024 Introduction to Laplace Transform Let f(x) be defined for 0 ≤ x < ∞ and let s denote an arbitrary real variable. The Laplace transform of f (x), designated by either L or F (s), is R∞ L {f (x)}=F (s)= 0 e−sx f (x)dx for all values of s for which the improper integral converges. Convergence occur when the limit RR limR−→∞ 0 e−sx f (x)dx exists. 1 Formulas used Prove that If L {f (x)} = F (s), L {g(x)} = G(s), then for any two constants c1 and c2 L {c1 f (x) + c2 g(x)} = c1 L {f (x)} + c2 L {g(x)} Proof L {c1 f (x) + c2 g(x)} = R∞ e−sx [c1 f (x) + c2 g(x)] dx R∞ R∞ L {c1 f (x) + c2 g(x)} = 0 e−sx [c1 f (x)] dx + 0 e−sx [c2 g(x)] dx R∞ R∞ L {c1 f (x) + c2 g(x)} = c1 0 e−sx f (x) dx + c2 0 e−sx g(x) dx 0 L {c1 f (x) + c2 g(x)} = c1 L {f (x)} + c2 L {g(x)} Also L {x sin ax} = L {x cos ax} = 2as (s2 +a2 )2 s2 −a2 (s2 +a2 )2 (s > 0) (s > 0)} Find L {x sin ax + x cos ax} Solution Using the formula L {c1 f (x) + c2 g(x)} = c1 L {f (x)} + c2 L {g(x)} Putting 2 c1 = c2 = 1, f (x) = x sin ax, g(x) = x cos ax L {x sin ax + x cos ax} = L {x sin ax} + L {x cos ax} Putting L {x sin ax} = 2as (s2 +a2 )2 (s > 0) L {x cos ax} = s2 −a2 (s2 +a2 )2 (s > 0)} L {x sin ax + x cos ax} = (s2 +a2 )2 L {x sin ax + x cos ax} = s2 +2as−a2 (s2 +a2 )2 2as Applications Find area under the curve y = e−7x (x sin 3x + x cos 3x) Solution Here s = 7, a = 3 2 Area= 7 +2×3×7−32 (72 +32 )2 = 41 1682 3 + s2 −a2 (s2 +a2 )2 (s > 0)