Question

4. Using the above properties, calculate the Laplace transform of the following func- tions, and indicate in each case their domain. • f(x) = 2 + x - 4x2 + 3x3 • f(0) = xer, (a € R) • f(x) = rear, (a € R, b> -1) • f(x) = 3e2x + x²e- 223" f(x) = cos(ax), (a € R) • f(x) = e-a sen(bx), (a, b e R) • f(x) = cosa 2

Answer 1

1

4 By definition of Laplace trang tom of f(x) L{ flai] - J * & tal de f (x) dx = f(p) --- given f (x) 2+x- 4x + 3x² ni { th} = L{} pht L {f(x)} = 2L81} + {2} - 4 L{x} + 3 L{x}} 2 4x 21 331 f(e) P P3 p4 f(p) 18 2 Its domain #8 PER-{o} f(x) = x x L{ t'f(t)= k) LAU P-a P-a L{x} = (-1) Lax (p-a) FP) PER-{a} (P-a)²

2

:) f(x)= 302t + Ret_ het P-2 Pt1 Lex} = Lf flu)}- 3446 27+ L x x 7 - 2 L2 2 } 1 P-2 dp² P+1 + +2 3 (-6) (P-1) 4 (P+)3 2 fle) 3. P-2 + PER_{2,-1, 1} (P+)3 (P-14 Its Domain ng f(x) = Cosax {1x)} { Corax} f(p) P p² + a² PER / It's domain R

3

vi) f(x) - é at sinbx) on Laplace By frost shifting property transform If L { FL+ }} = f (P) then at f(t)}- & (P-a) Now { sin be}= = f(e) p²+6² b Lead sinbx} = PE R (p+b) + b It's domain 2 vii) f(x) cos²x Cosar- 2 colet It Cosad L {it Caser} f(e) = {] .: It's domain & R- {}