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

Laplace Transform of x sin ax + 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)} L {sin ax} = a s2 +a2 (s > 0) L {cos ax} = s s2 +a2 (s > 0) Find L {x sin ax + cosax} Solution Using the formula 2 L {c1 f (x) + c2 g(x)} = c1 L {f (x)} + c2 L {g(x)} Putting c1 = c2 = 1, f (x) = xsinax, g(x) = cosax L {xsinax + cosax} = L {xsinax} + L {cosax} Putting L {x sin ax} = L {cos ax} = 2as (s2 +a2 )2 s s2 +a2 (s > 0) (s > 0) L {xsinax + cosax} = 2as (s2 +a2 )2 L {xsinax + cosax} = 2as+s(s2 +a2 ) (s2 +a2 )2 L {xsinax + cosax} = s(s2 +a2 +2a) (s2 +a2 )2 + Applications Find area under the curve y = e−7x (5sin5x + cos5x) Solution Here s = 7, a = 5 2 2 +2×5) Area= 7(7(7+5 = 2 +52 )2 147 1369 3 s (s2 +a2 ) (s > 0)