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- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.03%3A_Solution_of_Initial_Value_ProblemsThis section applies the Laplace transform to solve initial value problems for constant coefficient second order differential equations on (0,∞).
- https://math.libretexts.org/Under_Construction/Purgatory/Differential_Equations_and_Linear_Algebra_(Zook)/07%3A_Laplace_Transforms/7.03%3A_Solution_of_Initial_Value_ProblemsThis section applies the Laplace transform to solve initial value problems for constant coefficient second order differential equations on (0,∞).
- https://math.libretexts.org/Courses/Reedley_College/Differential_Equations_and_Linear_Algebra_(Zook)/13%3A_Laplace_Transforms/13.01%3A_Introduction_to_the_Laplace_TransformThis section defines the Laplace transform and develops its properties.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/09%3A_Laplace_Transforms/9.03%3A_Solution_of_Initial_Value_Problems\[{\cal L}\left(\sin(\omega t)\right)={\cal L}\left({-1\over \omega}{d\over dt}\cos(\omega t)\right)={-1\over \omega}{\cal L}\left({d\over dt}\cos(\omega t)\right)={-1\over \omega}\left(s{\cal L}(\cos...L(sin(ωt))=L(−1ωddtcos(ωt))=−1ωL(ddtcos(ωt))=−1ω(sL(cos(ωt)−cos(0))=−1ω(sss2+ω2−1)=−1ω−ω2s2+ω2=ωs2+ω2. L(f″
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.01%3A_Introduction_to_the_Laplace_TransformThis section defines the Laplace transform and develops its properties.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus_II__Integral_Calculus_._Lockman_Spring_2024/03%3A_Techniques_of_Integration/3.07%3A_Improper_IntegralsThis section covers improper integrals, focusing on integrals with infinite limits or integrands with infinite discontinuities. It explains how to evaluate these integrals by taking limits and determi...This section covers improper integrals, focusing on integrals with infinite limits or integrands with infinite discontinuities. It explains how to evaluate these integrals by taking limits and determining whether they converge or diverge. The section provides examples of both types of improper integrals, illustrating the steps needed to handle these situations effectively in calculus.
- https://math.libretexts.org/Courses/Chabot_College/Math_4%3A_Differential_Equations_(Dinh)/06%3A_Laplace_TransformsIn this Chapter we study the method of Laplace transforms, which illustrates one of the basic problem solving techniques in mathematics: transform a difficult problem into an easier one, solve the lat...In this Chapter we study the method of Laplace transforms, which illustrates one of the basic problem solving techniques in mathematics: transform a difficult problem into an easier one, solve the latter, and then use its solution to obtain a solution of the original problem. The method discussed here transforms an initial value problem for a constant coefficient equation into an algebraic equation whose solution can then be used to solve the initial value problem.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/09%3A_Laplace_Transforms/9.01%3A_Introduction_to_the_Laplace_Transform\[\begin{aligned}F(s)=\int_0^\infty e^{-st}(t)dt&=\lim_{T\to\infty}\int_0^T te^{-st}dt\\&=\lim_{T\to\infty}\left(-{te^{-st}\over s}\bigg|_0^T +{1\over s}\int_0^T e^{-st}dt\right)\\& =\lim_{T\to\infty}...\begin{aligned}F(s)=\int_0^\infty e^{-st}(t)dt&=\lim_{T\to\infty}\int_0^T te^{-st}dt\\&=\lim_{T\to\infty}\left(-{te^{-st}\over s}\bigg|_0^T +{1\over s}\int_0^T e^{-st}dt\right)\\& =\lim_{T\to\infty}\left({-Te^{-sT}\over s}+0-{e^{-st}\over s^2}\bigg|_0^T\right)\\&=\lim_{T\to\infty}\left({-Te^{-sT}\over s}-{e^{-sT}\over s^2}+{1\over s^2}\right)\\&=\left\{\begin{array}{rr} {1\over s^2}, & s>0,\\ \infty, & s<0. \end{array}\right.\end{aligned}\nonumber
- https://math.libretexts.org/Courses/Community_College_of_Denver/MAT_2562_Differential_Equations_with_Linear_Algebra/08%3A_Laplace_Transforms/8.01%3A_Introduction_to_the_Laplace_TransformThis section defines the Laplace transform and develops its properties.
- https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/08%3A_Laplace_Transforms/8.03%3A_Solution_of_Initial_Value_ProblemsThis section applies the Laplace transform to solve initial value problems for constant coefficient second order differential equations on (0,∞).
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_401%3A_Calculus_II_-_Integral_Calculus_Lecture_Notes_(Simpson)/02%3A_Techniques_of_Integration/2.07%3A_Improper_IntegralsIn this section, we define integrals over an infinite interval as well as integrals of functions containing a discontinuity on the interval. Integrals of these types are called improper integrals. We ...In this section, we define integrals over an infinite interval as well as integrals of functions containing a discontinuity on the interval. Integrals of these types are called improper integrals. We examine several techniques for evaluating improper integrals, all of which involve taking limits.