Laplace Transform Solved Problems - Univerzita Karlova
Laplace Transformsolved problemsPavel PyrihMay 24, 2012( public domain )Acknowledgement. The following problems were solved using my own procedurein a program Maple V, release 5, using commands fromBent E. Petersen: Laplace Transform in Maplehttp://people.oregonstate.edu/ peterseb/mth256/docs/256winter2001 laplace.pdfAll possible errors are my faults.1Solving equations using the Laplace transformTheorem.(Lerch) If two functions have the same integral transform then theyare equal almost everywhere.This is the right key to the following problems.Notation.(Dirac & Heaviside) The Dirac unit impuls function will be denotedby δ(t). The Heaviside step function will be denoted by u(t).1
1.1Problem.Using the Laplace transform find the solution for the following equation y(t) 3 2 t twith initial conditionsy(0) 0Dy(0) 0Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains Y(s) y(0) 311 2 2ssFrom this equation we solve Y (s)y(0) s2 3 s 2s3and invert it using the inverse Laplace transform and the same tables again andobtain t2 3 t y(0)With the initial conditions incorporated we obtain a solution in the form t2 3 tWithout the Laplace transform we can obtain this general solutiony(t) t2 3 t C1Info.polynomialComment.elementary2
1.2Problem.Using the Laplace transform find the solution for the following equation y(t) e( 3 t) twith initial conditionsy(0) 4Dy(0) 0Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains Y(s) y(0) 1s 3From this equation we solve Y (s)y(0) s 3 y(0) 1s (s 3)and invert it using the inverse Laplace transform and the same tables again andobtain11 y(0) e( 3 t)33With the initial conditions incorporated we obtain a solution in the form13 1 ( 3 t) e33Without the Laplace transform we can obtain this general solution1y(t) e( 3 t) C13Info.exponential functionComment.elementary3
1.3Problem.Using the Laplace transform find the solution for the following equation( y(t)) y(t) f(t) twith initial conditionsy(0) aDy(0) bHint.convolutionSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains Y(s) y(0) Y(s) laplace(f(t), t, s)From this equation we solve Y (s)y(0) laplace(f(t), t, s)s 1and invert it using the inverse Laplace transform and the same tables again andobtainZ ty(0) e( t) f( U1 ) e( t U1 ) d U10With the initial conditions incorporated we obtain a solution in the formZ ta e( t) f( U1 ) e( t U1 ) d U10Without the Laplace transform we can obtain this general solutionZy(t) e( t) f(t) et dt e( t) C1Info.exp convolutionComment.advanced4
1.4Problem.Using the Laplace transform find the solution for the following equation( y(t)) y(t) et twith initial conditionsy(0) 1Dy(0) 0Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains Y(s) y(0) Y(s) 1s 1From this equation we solve Y (s)y(0) s y(0) 1s2 1and invert it using the inverse Laplace transform and the same tables again andobtain1 t1e y(0) e( t) e( t)22With the initial conditions incorporated we obtain a solution in the form1 t 1 ( t)e e22Without the Laplace transform we can obtain this general solutiony(t) 1 te e( t) C12Info.exponential functionComment.elementary5
1.5Problem.Using the Laplace transform find the solution for the following equation( y(t)) 5 y(t) 0 twith initial conditionsy(0) 2Dy(0) bHint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains Y(s) y(0) 5 Y(s) 0From this equation we solve Y (s)y(0)s 5and invert it using the inverse Laplace transform and the same tables again andobtainy(0) e(5 t)With the initial conditions incorporated we obtain a solution in the form2 e(5 t)Without the Laplace transform we can obtain this general solutiony(t) C1 e(5 t)Info.exponential functionComment.elementary6
1.6Problem.Using the Laplace transform find the solution for the following equation( y(t)) 5 y(t) e(5 t) twith initial conditionsy(0) 0Dy(0) bHint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains Y(s) y(0) 5 Y(s) 1s 5From this equation we solve Y (s)y(0) s 5 y(0) 1s2 10 s 25and invert it using the inverse Laplace transform and the same tables again andobtaint e(5 t) y(0) e(5 t)With the initial conditions incorporated we obtain a solution in the formt e(5 t)Without the Laplace transform we can obtain this general solutiony(t) t e(5 t) C1 e(5 t)Info.exponential functionComment.elementary7
1.7Problem.Using the Laplace transform find the solution for the following equation( y(t)) 5 y(t) e(5 t) twith initial conditionsy(0) 2Dy(0) bHint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains Y(s) y(0) 5 Y(s) 1s 5From this equation we solve Y (s)y(0) s 5 y(0) 1s2 10 s 25and invert it using the inverse Laplace transform and the same tables again andobtaint e(5 t) y(0) e(5 t)With the initial conditions incorporated we obtain a solution in the formt e(5 t) 2 e(5 t)Without the Laplace transform we can obtain this general solutiony(t) t e(5 t) C1 e(5 t)Info.exponential functionComment.elementary8
1.8Problem.Using the Laplace transform find the solution for the following equation 2y(t) f(t) t2with initial conditionsy(0) aDy(0) bHint.convolutionSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) laplace(f(t), t, s)From this equation we solve Y (s)y(0) s D(y)(0) laplace(f(t), t, s)s2and invert it using the inverse Laplace transform and the same tables again andobtainZ ty(0) D(y)(0) t f( U1 ) (t U1 ) d U10With the initial conditions incorporated we obtain a solution in the formZ ta bt f( U1 ) (t U1 ) d U10Without the Laplace transform we can obtain this general solutionZ Zy(t) f(t) dt C1 dt C2Info.convolutionComment.advanced9
1.9Problem.Using the Laplace transform find the solution for the following equation 2y(t) 1 t t2with initial conditionsy(0) 0Dy(0) 0Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 11 2s sFrom this equation we solve Y (s)s3 y(0) D(y)(0) s2 s 1s4and invert it using the inverse Laplace transform and the same tables again andobtain11 t3 t2 D(y)(0) t y(0)62With the initial conditions incorporated we obtain a solution in the form11 t3 t262Without the Laplace transform we can obtain this general solutiony(t) 1 2 1 3t t C1 t C226Info.polynomialComment.elementary10
1.10Problem.Using the Laplace transform find the solution for the following equation 2 y(t) 2 ( y(t)) y(t) t2 twith initial conditionsy(0) 3Dy(0) 6Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 2 s Y(s) 2 y(0) Y(s)From this equation we solve Y (s)y(0) s D(y)(0) 2 y(0)s2 2 s 1and invert it using the inverse Laplace transform and the same tables again andobtain 11 t e 2 D(y)(0) sinh( 2 t) et y(0) 2 sinh( 2 t) et y(0) cosh( 2 t)22With the initial conditions incorporated we obtain a solution in the form 3 t e 2 sinh( 2 t) 3 et cosh( 2 t)2Without the Laplace transform we can obtain this general solutiony(t) C1 e(( 2 1) t) C2 e( (Info.3 e(2 t)Comment.elementary11 2 1) t)
1.11Problem.Using the Laplace transform find the solution for the following equation 2y(t) 3 2 t t2with initial conditionsy(0) aDy(0) bHint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 311 2 2ssFrom this equation we solve Y (s)s3 y(0) D(y)(0) s2 3 s 2s4and invert it using the inverse Laplace transform and the same tables again andobtain1 3 3 2t t D(y)(0) t y(0)32With the initial conditions incorporated we obtain a solution in the form1 3 3 2t t bt a32Without the Laplace transform we can obtain this general solutiony(t) 3 2 1 3t t C1 t C223Info.polynomialComment.elementary12
1.12Problem.Using the Laplace transform find the solution for the following equation 2y(t) 3 2 t t2with initial conditionsy(0) aDy(0) bHint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 311 2 2ssFrom this equation we solve Y (s)s3 y(0) D(y)(0) s2 3 s 2s4and invert it using the inverse Laplace transform and the same tables again andobtain13 t3 t2 D(y)(0) t y(0)32With the initial conditions incorporated we obtain a solution in the form13 t3 t2 b t a32Without the Laplace transform we can obtain this general solutiony(t) 3 2 1 3t t C1 t C223Info.polynomialComment.elementary13
1.13Problem.Using the Laplace transform find the solution for the following equation( 2y(t)) 16 y(t) 5 δ(t 1) t2with initial conditionsy(0) 0Dy(0) 0Hint.care!Solution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 16 Y(s) 5 e( s)From this equation we solve Y (s)y(0) s D(y)(0) 5 e( s)s2 16and invert it using the inverse Laplace transform and the same tables again andobtainy(0) cos(4 t) 15D(y)(0) sin(4 t) u(t 1) sin(4 t 4)44With the initial conditions incorporated we obtain a solution in the form5u(t 1) sin(4 t 4)4Without the Laplace transform we can obtain this general solution55cos(4) u(t 1) sin(4 t) sin(4) u(t 1) cos(4 t) C1 sin(4 t)44 C2 cos(4 t)y(t) Info.u and trig functionsComment.advanced14
1.14Problem.Using the Laplace transform find the solution for the following equation 2y(t)) 16 y(t) 16 u(t 3) 16 t2with initial conditions(y(0) 0Dy(0) 0Hint.care!Solution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 16 Y(s) 16e( 3 s)1 16ssFrom this equation we solve Y (s)y(0) s2 D(y)(0) s 16 e( 3 s) 16s (s2 16)and invert it using the inverse Laplace transform and the same tables again andobtainy(0) cos(4 t) 1D(y)(0) sin(4 t) u(t 3) u(t 3) cos(4 t 12) 14 cos(4 t)With the initial conditions incorporated we obtain a solution in the form 1 u(t 3) u(t 3) cos(4 t 12) cos(4 t)Without the Laplace transform we can obtain this general solutiony(t) (u(t 3) sin(4 t) u(t 3) sin(12) sin(4 t)) sin(4 t) (cos(4 t) u(t 3) u(t 3) cos(12) cos(4 t)) cos(4 t) C1 sin(4 t) C2 cos(4 t)Info.u and trig functionsComment.advanced15
1.15Problem.Using the Laplace transform find the solution for the following equation( 2 y(t)) 2 ( y(t)) 2 y(t) 0 t2 twith initial conditionsy(0) 1Dy(0) 1Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 2 s Y(s) 2 y(0) 2 Y(s) 0From this equation we solve Y (s)y(0) s D(y)(0) 2 y(0)s2 2 s 2and invert it using the inverse Laplace transform and the same tables again andobtaine( t) D(y)(0) sin(t) e( t) y(0) sin(t) e( t) y(0) cos(t)With the initial conditions incorporated we obtain a solution in the forme( t) cos(t)Without the Laplace transform we can obtain this general solutiony(t) C1 e( t) sin(t) C2 e( t) cos(t)Info.e( t) cos(t)Comment.standard16
1.16Problem.Using the Laplace transform find the solution for the following equation( 2y(t)) 2 ( y(t)) 2 y(t) f(t) t2 twith initial conditionsy(0) 0Dy(0) 0Hint.convolutionSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 2 s Y(s) 2 y(0) 2 Y(s) laplace(f(t), t, s)From this equation we solve Y (s)y(0) s D(y)(0) 2 y(0) laplace(f(t), t, s)s2 2 s 2and invert it using the inverse Laplace transform and the same tables again andobtaine( t) y(0) cos(t) e( t) y(0) sin(t) e( t) D(y)(0) sin(t)Z t f( U1 ) e( t U1 ) sin( t U1 ) d U10With the initial conditions incorporated we obtain a solution in the formZ t f( U1 ) e( t U1 ) sin( t U1 ) d U10Without the Laplace transform we can obtain this general solutionZZy(t) sin(t) f(t) et dt e( t) cos(t) cos(t) f(t) et dt e( t) sin(t) C1 e( t) cos(t) C2 e( t) sin(t)Info.sin convolutionComment.standard17
1.17Problem.Using the Laplace transform find the solution for the following equation( 2y(t)) 4 y(t) 0 t2with initial conditionsy(0) 2Dy(0) 2Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 4 Y(s) 0From this equation we solve Y (s)y(0) s D(y)(0)s2 4and invert it using the inverse Laplace transform and the same tables again andobtain1D(y)(0) sin(2 t) y(0) cos(2 t)2With the initial conditions incorporated we obtain a solution in the formsin(2 t) 2 cos(2 t)Without the Laplace transform we can obtain this general solutiony(t) C1 cos(2 t) C2 sin(2 t)Info.trig functionsComment.elementary18
1.18Problem.Using the Laplace transform find the solution for the following equation( 2y(t)) 4 y(t) 6 y(t) t2with initial conditionsy(0) 6Dy(0) 0Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 4 Y(s) 6 Y(s)From this equation we solve Y (s)y(0) s D(y)(0)s2 2and invert it using the inverse Laplace transform and the same tables again andobtain 1 2 D(y)(0) sinh( 2 t) y(0) cosh( 2 t)2With the initial conditions incorporated we obtain a solution in the form 6 cosh( 2 t)Without the Laplace transform we can obtain this general solution y(t) C1 sinh( 2 t) C2 cosh( 2 t)Info.sinh coshComment.standard19
1.19Problem.Using the Laplace transform find the solution for the following equation( 2y(t)) 4 y(t) cos(t) t2with initial conditionsy(0) aDy(0) bHint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 4 Y(s) ss2 1From this equation we solve Y (s)s3 y(0) y(0) s D(y)(0) s2 D(y)(0) ss4 5 s2 4and invert it using the inverse Laplace transform and the same tables again andobtain111 cos(2 t) y(0) cos(2 t) D(y)(0) sin(2 t) cos(t)323With the initial conditions incorporated we obtain a solution in the form111 cos(2 t) a cos(2 t) b sin(2 t) cos(t)323Without the Laplace transform we can obtain this general solution1111cos(3 t) cos(t)) cos(2 t) ( sin(t) sin(3 t)) sin(2 t) C1 cos(2 t)124412 C2 sin(2 t)y(t) (Info.trig functionsComment.standard20
1.20Problem.Using the Laplace transform find the solution for the following equation( 2 y(t)) 9 ( y(t)) 20 y(t) f(t) t2 twith initial conditionsy(0) 0Dy(0) 0Hint.convolutionSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 9 s Y(s) 9 y(0) 20 Y(s) laplace(f(t), t, s)From this equation we solve Y (s)y(0) s D(y)(0) 9 y(0) laplace(f(t), t, s)s2 9 s 20and invert it using the inverse Laplace transform and the same tables again andobtain 4 y(0) e( 5 t) 5 y(0) e( 4 t) D(y)(0) e( 5 t) D(y)(0) e( 4 t)Z tZ tf( U2 ) e( 4 t 4 U2 ) d U2 f( U1 ) e( 5 t 5 U1 ) d U1 00With the initial conditions incorporated we obtain a solution in the formZ tZ t f( U1 ) e( 5 t 5 U1 ) d U1 f( U2 ) e( 4 t 4 U2 ) d U200Without the Laplace transform we can obtain this general solutionZZ(4 t)(5 t)y(t) ( f(t) edt e f(t) e(5 t) dt e(4 t) ) e( 9 t) C1 e( 4 t) C2 e( 5 t)Info.exp convolutionComment.standard21
1.21Problem.Using the Laplace transform find the solution for the following equation( 2y(t)) 9 y(t) 0 t2with initial conditionsy(0) 3Dy(0) 5Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) 9 Y(s) 0From this equation we solve Y (s)y(0) s D(y)(0)s2 9and invert it using the inverse Laplace transform and the same tables again andobtain1D(y)(0) sin(3 t) y(0) cos(3 t)3With the initial conditions incorporated we obtain a solution in the form5 sin(3 t) 3 cos(3 t)3Without the Laplace transform we can obtain this general solutiony(t) C1 cos(3 t) C2 sin(3 t)Info.trig functionsComment.standard22
1.22Problem.Using the Laplace transform find the solution for the following equation( 2y(t)) y(t) 0 t2with initial conditionsy(0) 0Dy(0) 1Hint.no hintSolution.We denote Y (s) L(y)(t) the Laplace transform Y (s) of y(t). We perform theLaplace transform for both sides of the given equation. For particular functionswe use tables of the Laplace transforms and obtains (s Y(s) y(0)) D(y)(0) Y(s) 0From this equation we solve Y (s)y(0) s D(y)(0)s2 1and invert it using the inverse Laplace transform and the same tables again andobtainy(0) cos(t) D(y)(0) sin(t)With the initial conditions incorporated we obtain a solution in the formsin(t)Without the Laplace transform we can obtain this general solutiony(t) C1 cos(t) C2 sin(t)Info.trig functionsComment.standard23
1.23Problem.Using the Laplace transform find the solution for the following equation( 2 y(t))
Laplace Transform solved problems Pavel Pyrih May 24, 2012 ( public domain ) Acknowledgement.The following problems were solved using my own procedure
Using the Laplace Transform, differential equations can be solved algebraically. 2. We can use pole/zero diagrams from the Laplace Transform to determine the frequency response of a system and whether or not the system is stable. 3. We can tra
Laplace Transform in Engineering Analysis Laplace transforms is a mathematical operation that is used to “transform” a variable (such as x, or y, or z, or t)to a parameter (s)- transform ONE variable at time. Mathematically, it can be expressed as: L f t e st f t dt F s t 0 (5.1) In a layman’s term, Laplace transform is used
Introduction to Laplace transform methods Page 1 A Short Introduction to Laplace Transform Methods (tbco, 3/16/2017) 1. Laplace transforms a) Definition: Given: ( ) Process: ℒ( ) ( ) 0 ̂( ) Result: ̂( ), a function of the “Laplace tr
Nov 27, 2014 · If needed we can find the inverse Laplace transform, which gives us the solution back in "t-space". Definition of Laplace Transform Let be a given function which is defined for . If there exists a function so that , Then is called the Laplace Transform of , and will be denoted by
Objectives: Objectives: Calculate the Laplace transform of common functions using the definition and the Laplace transform tables Laplace-transform a circuit, including components with non-zero initial conditions. Analyze a circuit in the s-domain Check your s-domain answers using the initial value theorem (IVT) and final value theorem (FVT)
Dec 05, 2014 · The Laplace transform of the convolution of fand gis equal to the product of the Laplace transformations of fand g, i.e. L[fg]( ) F( ) G( ) In other words, the Laplace transform \turns convolution into multiplication." 1.4.3 Derivative rule First
No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain (t) Transform domain (s) Original DE & IVP Algebraic equation for the Laplace transform Laplace transform of the solu
mastery of it : writing . In a recent survey, academic staff at the University identified the interrelated skills of essay-writing and reasoning as the two most important skills for success in higher education; when asked which skills students most often lacked, essay-writing was again at the top of their list. Needless to say, writing ability is also highly prized by employers. The purpose of .
