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1. FORMULAS TO REMEMBER L(eat f(t)) = F(s-a); c(rt-c)u₂(t)) = ecif(t)); £¹(F(s-a)) = eªt f(t) where f(t) = £¹(F(s)) ¹(e-F(s)) = f(t-c)u(t) where f(t) = ¹ (F(s)) L(f(t)8(t-c)) = f(c)e- f(t) *g(t) = [*f{r}g{t= v)dr = [*g(1)/(1-1) (F(s)G(s)) = f(t)-g(t) f(t-1)dr b) L(8(t-c)) = e-s; L(f(t) g(t)) = F(s)G(s); Find the Laplace of the following functions. a) J(t)=et et cosh(at) K(t)=sinre-+*cos(t-t)dt - İsinr

1. FORMULAS TO REMEMBER L(eat f(t)) = F(s-a); c(rt-c)u₂(t)) = ecif(t)); £¹(F(s-a)) = eªt f(t) where f(t) = £¹(F(s)) ¹(e-F(s)) = f(t-c)u(t) where f(t) = ¹ (F(s)) L(f(t)8(t-c)) = f(c)e- f(t) *g(t) = [*f{r}g{t= v)dr = [*g(1)/(1-1) (F(s)G(s)) = f(t)-g(t) f(t-1)dr b) L(8(t-c)) = e-s; L(f(t) g(t)) = F(s)G(s); Find the Laplace of the following functions. a) J(t)=et et cosh(at) K(t)=sinre-+*cos(t-t)dt - İsinr

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Plz do both parts and take a thumb up Plz do both parts and take a thumb up
**Formulas to Remember** 1. \( \mathcal{L}(e^{at}f(t)) = F(s - a); \quad \mathcal{L}^{-1}(F(s - a)) = e^{at}f(t) \) where \( f(t) = \mathcal{L}^{-1}(F(s)) \) 2. \( \mathcal{L}( f(t - c)u_c(t) ) = e^{-cs}L(f(t)); \quad \mathcal{L}^{-1}( e^{-cs}F(s) ) = f(t-c)u_c(t) \) where \( f(t) = \mathcal{L}^{-1}(F(s)) \) 3. \( \mathcal{L}( \delta(t - c) ) = e^{-cs} \) 4. \( \mathcal{L}( f(t)\delta(t - c) ) = f(c)e^{-cs} \) 5. \( \mathcal{L}( f(t) \ast g(t) ) = F(s)G(s); \quad \mathcal{L}^{-1}( F(s)G(s) ) = f(t) \ast g(t) \) 6. \( f(t) + g(t) = \int_0^t f(t - \tau) g(\tau) d\tau = \int_0^t g(t - \tau) f(\tau) d\tau \) --- 1. Find the Laplace of the following functions. a) \( j(t) = e^{2t} + e^t \cosh(at) \) b) \( K(t) = \int_0^t \sin \tau \, e^{t-\tau} \cos(t - \tau) d\tau \)
Transcribed Image Text:**Formulas to Remember** 1. \( \mathcal{L}(e^{at}f(t)) = F(s - a); \quad \mathcal{L}^{-1}(F(s - a)) = e^{at}f(t) \) where \( f(t) = \mathcal{L}^{-1}(F(s)) \) 2. \( \mathcal{L}( f(t - c)u_c(t) ) = e^{-cs}L(f(t)); \quad \mathcal{L}^{-1}( e^{-cs}F(s) ) = f(t-c)u_c(t) \) where \( f(t) = \mathcal{L}^{-1}(F(s)) \) 3. \( \mathcal{L}( \delta(t - c) ) = e^{-cs} \) 4. \( \mathcal{L}( f(t)\delta(t - c) ) = f(c)e^{-cs} \) 5. \( \mathcal{L}( f(t) \ast g(t) ) = F(s)G(s); \quad \mathcal{L}^{-1}( F(s)G(s) ) = f(t) \ast g(t) \) 6. \( f(t) + g(t) = \int_0^t f(t - \tau) g(\tau) d\tau = \int_0^t g(t - \tau) f(\tau) d\tau \) --- 1. Find the Laplace of the following functions. a) \( j(t) = e^{2t} + e^t \cosh(at) \) b) \( K(t) = \int_0^t \sin \tau \, e^{t-\tau} \cos(t - \tau) d\tau \)
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