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  • https://math.libretexts.org/Courses/East_Tennesee_State_University/Book%3A_Differential_Equations_for_Engineers_(Lebl)_Cintron_Copy/6%3A_The_Laplace_Transform/6.1%3A_The_Laplace_Transform
    The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If th...The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If the algebraic equation can be solved, applying the inverse transform gives us our desired solution.
  • https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/6%3A_Power_Series_and_Laplace_Transforms/6.8%3A_Step_Functions
    In this discussion, we will investigate piecewise defined functions and their Laplace Transforms. We start with the fundamental piecewise defined function, the Heaviside function.
  • https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Yet_Another_Calculus_Text__A_Short_Introduction_with_Infinitesimals_(Sloughter)/01%3A_Derivatives/1.04%3A_Continuous_Functions
    \[ U(t)=\left\{\begin{array}{ll}{0,} & {\text { if } t<0,} \\ {1,} & {\text { if } 0 \leq t \leq 1,} \\ {0,} & {\text { if } t>1,}\end{array}\right. \] is continuous from the right at \(t=0\) and cont...\[ U(t)=\left\{\begin{array}{ll}{0,} & {\text { if } t<0,} \\ {1,} & {\text { if } 0 \leq t \leq 1,} \\ {0,} & {\text { if } t>1,}\end{array}\right. \] is continuous from the right at \(t=0\) and continuous from the left at \(t=1,\) but not continuous at either \(t=0\) or \(t=1 .\) See Figure \(1.4 .2 .\) Given real numbers \(a\) and \(b,\) we let \[ [a, b]=\{x | x \text { is a real number and } a \leq x \leq b\} , \] \[ [a, \infty)=\{x | x \text { is a real number and } x \geq a\} , \] and \[ …
  • https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/6%3A_The_Laplace_Transform/6.1%3A_The_Laplace_Transform
    The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If th...The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If the algebraic equation can be solved, applying the inverse transform gives us our desired solution.
  • https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Ordinary_Differential_Equations/6%3A_Power_Series_and_Laplace_Transforms/6.9%3A_Discontinuous_Forcing
    In the last section we looked at the Heaviside function its Laplace transform. Now we will use this tool to solve differential equations.
  • https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/08%3A_The_Laplace_Transform/8.01%3A_The_Laplace_Transform
    The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If th...The Laplace transform turns out to be a very efficient method to solve certain ODE problems. In particular, the transform can take a differential equation and turn it into an algebraic equation. If the algebraic equation can be solved, applying the inverse transform gives us our desired solution.
  • https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.04%3A_The_Dirac_Delta_Function
    The Dirac delta function, δ(x) this is one example of what is known as a generalized function, or a distribution. Dirac had introduced this function in the 1930′s in his study of quantum mechanics a...The Dirac delta function, δ(x) this is one example of what is known as a generalized function, or a distribution. Dirac had introduced this function in the 1930′s in his study of quantum mechanics as a useful tool. It was later studied in a general theory of distributions and found to be more than a simple tool used by physicists. The Dirac delta function, as any distribution, only makes sense under an integral.
  • https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.02%3A_Transforms_of_Derivatives_and_ODEs
    The procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into ...The procedure for linear constant coefficient equations is as follows. We take an ordinary differential equation in the time variable t . We apply the Laplace transform to transform the equation into an algebraic (non differential) equation in the frequency domain.  We solve the equation for X(s) . Then taking the inverse transform, if possible, we find x(t). Unfortunately, not every function has a Laplace transform, not every equation can be solved in this manner.

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