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Ejercicios 7.3 Problema 21 Dennis G. ZILL ED 9na Ed. Transformada de Laplace ED PVI
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Ejercicios 7.3 Problema 21 Dennis G. ZILL ED 9na Ed. Transformada de Laplace ED PVI

How to Solve an Initial Value Problem Using Laplace Transform

Introduction to the Problem

  • The video introduces the topic of solving a differential equation with an initial value problem using the Laplace transform.
  • It emphasizes the need to apply the Laplace transform to convert the differential equation into an algebraic equation.

Steps in Applying Laplace Transform

  • The speaker explains that they will separate terms and apply the Laplace transform, focusing on constants as factors similar to integrals and derivatives.
  • They mention obtaining transforms for specific functions, particularly noting how to handle derivatives during this process.

Deriving Transforms

  • The first derivative is discussed, where it’s noted that s cdot Y(s) - y(0) is used for transformation.
  • The speaker calculates the transform of an exponential function, emphasizing its formula and substituting values correctly.

Substituting Transforms into Equation

  • After obtaining necessary transforms, they substitute these back into their original equation involving Y(s) .
  • A rearrangement of terms leads them towards isolating Y(s) , preparing for further simplification.

Finalizing Expression and Inverse Transform

  • The expression for Y(s) is simplified down to a fraction which can be factored further.
  • They break down Y(s) , separating it into two fractions for easier handling during inverse transformation.

Application of Inverse Transform Techniques

  • The speaker discusses applying inverse transformations using known formulas, specifically addressing exponential functions.
  • They highlight using translation theorem techniques when evaluating inverse transforms related to shifts in variables.

Inverse Laplace Transform and Exponential Functions

Understanding the Inverse Laplace Transform

  • The discussion begins with the evaluation of the inverse Laplace transform, specifically focusing on a function evaluated at s - at . The speaker emphasizes performing a direct inverse transformation.
  • The inverse transform of 1/s-a is noted to be e^at , where in this case, a = -4 . Thus, it results in e^-4t .
  • The speaker references previous chapters or sections regarding the inverse transform of 1/s^2 , which is simply represented as the variable multiplied by time ( t ).
  • It is clarified that the result from the inverse transform of 1/s^2 will yield an expression involving both time and an exponential factor due to multiplication with e^-4t .
  • A comparison is made with a textbook solution for problem 21 from exercise group 7.3, confirming that the derived answer matches expectations: it includes terms like e^-4t + 2e^-400 , indicating rearrangement rather than alteration of values.

Conclusion and Engagement

Video description

Ecuaciones Diferenciales con problemas de calores en la frontera 9na Ed. Dennis G. Zill Capítulo 7. Transformada de Laplace Ejercicios 7.3. Propiedades Operacionales I Problema 21: En los problemas 21-30, use la transformada de Laplace para resolver el problema con valores iniciales 21. y^′+4y=e^(−4t), y(0)=2 Bienvenidos al Canal @AprendeconOscarValdez, SUSCRÍBETE Te comparto la lista de reproducción referente a este grupo de ejercicios y formulario: Formulario derivadas integrales y transformadas de Laplace https://youtu.be/7vOwzDeS1s0 Transformada de Laplace: https://youtube.com/playlist?list=PLrEts5JskOlH2XfVZbaxc_2uotQZ0YqqS Adjunto el link de la lista de reproducción Ecuaciones diferenciales: https://youtube.com/playlist?list=PLrEts5JskOlFt-3btb1j4cArJicItMKzz En la lista de reproducción podrás encontrar diversos temas un poco de teoría y sobre todo ejercicios resueltos Referencias Zill, D. G. (2018). Ecuaciones Diferenciales con problemas de valores de frontera (Novena ed.). México: Cengage Learning. Corresponde también a ediciones anteriores