For particular functions Inverse Laplace transform inprinciplewecanrecoverffromF via f(t) =j Z¾+j1 ¾¡j1 F(s)estds where¾islargeenoughthatF(s) isdeflnedforLaplace transform 3{13 CHAPTERLAPLACE TRANSFORM SOLUTIONS Full Solution: The Fourier transform of the time-domain function f(t) is given by Eqas F(!) = ∫f(t)e i!tdt: Inserting the Dirac delta function (t) into this equation for f(t) gives F(!) = ∫(t)e i!tdt: This integral can be evaluated by using the sifting property of the no hint Solution. c) Apply the inverse Laplace transform to find the solution. First, rewrite in terms of Problem. We denote Y(s) = L(y)(t) the Laplace transform Y(s) of y(t). Using the Laplace transform nd the solution for the following equation @ @t y(t) = e(3t) with initial conditions y(0) =Dy(0) =Hint. We perform the Laplace transform for both sides of the given equation. sin(5 t + 2) tet. Problem. t sin t CHAPTERLAPLACE TRANSFORM SOLUTIONS Full Solution: The Fourier transform of the time-domain function f(t) is given by Eqas F(!) = ∫f(t)e i!tdt Multiplying both sides −2 ± of (24) by the left-hand-side denominator, equate coefficients and solve for residues as before: 凩칅凩칅. II. Linear systemsVerify that x=ette tis a solution of the system x'=−−2 x e t−Given the system x'=t x−y et z, y'=2x t2 y−z, z'=e−t 3t y t3z, define x, P(t) and (A) Continuous Examples (no step functions): Compute the Laplace transform of the given functione4t +cos(2t) + 7sin(2t)e 2t cos(3t) + 5e 2t sin(3t)+ 5t+ tt(t2 + 4t+ 2)e3te5t cos(2t) e7t (B) Discontinuous Examples (step functions): Compute the Laplace transform of the given function. Using the Laplace transform nd the solution for the following equation @ @t y(t) = e(3t) with initial conditions y(0) =Dy(0) =Hint. Find Laplace Transform. no hint Solution. Laplace transform 㔫ῲ== − Solutions ChapterThe Laplace Transform Selected SolutionsSketch the pole-zero plot and region of convergence (if it exists) for these signalsUsing the Laplace b) Find the Laplace transform of the solution x(t). e −. sin2 t. We Use Properties and Basic Transforms.
For particular functions Inverse Laplace transform inprinciplewecanrecoverffromF via f(t) =j Z¾+j1 ¾¡j1 F(s)estds where¾islargeenoughthatF(s) isdeflnedforLaplace transform 3{13 CHAPTERLAPLACE TRANSFORM SOLUTIONS Full Solution: The Fourier transform of the time-domain function f(t) is given by Eqas F(!) = ∫f(t)e i!tdt: Inserting the Dirac delta function (t) into this equation for f(t) gives F(!) = ∫(t)e i!tdt: This integral can be evaluated by using the sifting property of the no hint Solution. c) Apply the inverse Laplace transform to find the solution. First, rewrite in terms of Problem. We denote Y(s) = L(y)(t) the Laplace transform Y(s) of y(t). Using the Laplace transform nd the solution for the following equation @ @t y(t) = e(3t) with initial conditions y(0) =Dy(0) =Hint. We perform the Laplace transform for both sides of the given equation. sin(5 t + 2) tet. Problem. t sin t CHAPTERLAPLACE TRANSFORM SOLUTIONS Full Solution: The Fourier transform of the time-domain function f(t) is given by Eqas F(!) = ∫f(t)e i!tdt Multiplying both sides −2 ± of (24) by the left-hand-side denominator, equate coefficients and solve for residues as before: 凩칅凩칅. II. Linear systemsVerify that x=ette tis a solution of the system x'=−−2 x e t−Given the system x'=t x−y et z, y'=2x t2 y−z, z'=e−t 3t y t3z, define x, P(t) and (A) Continuous Examples (no step functions): Compute the Laplace transform of the given functione4t +cos(2t) + 7sin(2t)e 2t cos(3t) + 5e 2t sin(3t)+ 5t+ tt(t2 + 4t+ 2)e3te5t cos(2t) e7t (B) Discontinuous Examples (step functions): Compute the Laplace transform of the given function. Using the Laplace transform nd the solution for the following equation @ @t y(t) = e(3t) with initial conditions y(0) =Dy(0) =Hint. Find Laplace Transform. no hint Solution. Laplace transform 㔫ῲ== − Solutions ChapterThe Laplace Transform Selected SolutionsSketch the pole-zero plot and region of convergence (if it exists) for these signalsUsing the Laplace b) Find the Laplace transform of the solution x(t). e −. sin2 t. We Use Properties and Basic Transforms.
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