Laplace Transform Fact Sheet - Applied Differential Equations | MTH 238, Study notes of Differential Equations

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Laplace Transform Fact Sheet 1. Transforms and Inverse Transforms of Elementary Functions a £{I}= bande '{t} =1 b. £40) = 2 and se {a} f, n= 1,23. c. Yet) = “ea 4 a L{sindt} = —k = sinkt nee e. L£fcoskt} = —S — and £- uot = coskt we stk f. £fsinhkty = = and zi = } = sinhit stk 9. £ {cosh kt} = 55 and t1{ 5 | = coshia 2. Transforming a Function Containing the Factor e” a £{e"AO} = [LO] 2 b. £-14F(s—a)} = £1GF(s)), gf = eM LLRs) 3. Transforming a Function Containing the Factor ¢”, » = 1,2,3,... a LER} = (1) [LMO}] 4. Transforming Derivatives f a £{/'()} = s£{4)} --A0) b. £47" ()} = PL} ~ 90) 0) 5. Transforming a Function Containing the Factor U(t- a) a. £{ft- aUlt- a)} = e PL GO} b. LIAHUG- ah} = e OLY + a)} c. £ Me F(s)} = UE-a) ZF}, , 6. Transforming the Dirac Delta Function L{6(t—a)} = eo ® and £- fe} = A(t - a) 7. Convolution Theorem a feg= [fenete -wdu = [,e@one- udu b. £{f0) * a0} = LAN} + £490} c. £71 {F(s)G(s)} = [ee FHL, [£1 {CG)}],_ du 8. Transforming a Periodic Function Ift) is periodic and P is the period then 24/()} = Po ; a is e f(t yd.