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Understanding Laplace Transforms and Their Applications in Differential Equations

Explore Laplace Transforms in B.Sc. II Year Mathematics. Find solutions to differential equations with algebraic methods. Learn about the relationship to Fourier Transform and how it helps characterize systems efficiently.

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Understanding Laplace Transforms and Their Applications in Differential Equations

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  1. Department of MathematicsYashwantraoChavanMahavidyalaya,Tuljapur Osmanabad Presented byProf. Anil D. Patil

  2. Laplace Transform B.Sc. II Year

  3. Find solution to differential equation using algebra • Relationship to Fourier Transform allows easy way to characterize systems • No need for convolution of input and differential equation solution • Useful with multiple processes in system Why use Laplace Transforms?

  4. Find differential equations that describe system • Obtain Laplace transform • Perform algebra to solve for output or variable of interest • Apply inverse transform to find solution How to use Laplace?

  5. What are Laplace transforms? • t is real, s is complex! • Inverse requires complex analysis to solve • Note “transform”: f(t)  F(s), where t is integrated and s is variable • Conversely F(s)  f(t), t is variable and s is integrated • Assumes f(t) = 0 for all t < 0

  6. Hard Way – do the integral let let let Evaluating F(s) = L{f(t)} Integrate by parts

  7. remember let Substituting, we get: let Evaluating F(s)=L{f(t)}- Hard Way It only gets worse…

  8. This is the easy way ... • Recognize a few different transforms • See table 2.3 on page 42 in textbook • Or see handout .... • Learn a few different properties • Do a little math Evaluating F(s) = L{f(t)}

  9. Table of selected Laplace Transforms

  10. More transforms

  11. Unit step function definition: Used in conjunction with f(t)  f(t)u(t) because of Laplace integral limits: Note on step functions in Laplace

  12. Linearity • Scaling in time • Time shift • “frequency” or s-plane shift • Multiplication by tn • Integration • Differentiation Properties of Laplace Transforms

  13. Properties: Linearity Example : Proof :

  14. Proof : Example : let Properties: Scaling in Time

  15. Example : Proof : let Properties: Time Shift

  16. Example : Proof : Properties: S-plane (frequency) shift

  17. Example : Proof : Properties: Multiplication by tn

  18. Thanks

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