LAPLACE TRANSFORMS

                          LAPLACE TRANSFORMS

                      What we know is not much. 
                What we don't know is enormous.
                                      - pierre Simon Laplace

  Laplace transforms is used to find the solution of ordinary differential equations of any orders. In 12 th standard we learnt, how to solve first order Differential Equations (chapter-10). In this we are going to see the easy ways to solve higher order Differential Equations.

  Laplace transforms are important for any engineer's mathematical toolbox as they make solving linear Ordinary Differential Equations (ODEs) and related initial value problems, as well as systems of linear ODEs, much easier . 

  Applications abound : electrical networks, springs, mixing problems, signal processing, and other areas of engineering and physics.



Steps for transforming a equation to Laplace transform:

Laplace Transform. Linearity.
First Shifting Theorem (s-Shifting)

Some Functions ƒ(t) and Their Laplace Transforms

s-Shifting: Replacing s by s-a in the Transform


EX 6.1 Laplace transforms answers pdf link -https://drive.google.com/file/d/1b5YhCdrCDm-s-MJQBQyFPC43U3A7a2TN/view?usp=drivesdk
In exercise, 6.1 question number 43 the final answer is cost and sint. (Not cosh t sinh t) just a careless mistake. Please correct it.
Question number 33.

Ex 6.2 Transforms of derivatives and integrals. ODEs . Solving every prb will takes long time, so I solved the only selected prbs which seems to be critical.
https://drive.google.com/file/d/1drAbIDvuCad0kuKagEvwmp7TlQnrKWwd/view?usp=drivesdk

Ex 6.3 Unit step function ( Heaviside Function) second shifting theorem.
https://drive.google.com/file/d/1rkc3TpyoujEHbmri5VRW8EB4-_Gi2Vta/view?usp=drivesdk
Ex 6.4 Short Impulses. Dirac's Delta Function . Partial Fractions. https://drive.google.com/file/d/1xHcLaxtEsUVn4iSg7dKrARMKYvP2EEYu/view?usp=drivesdk
Ex . 6.5 convolution- integral equations.https://drive.google.com/file/d/105K5MsZkFwC67V8FEfkHx_m87i1bZpi5/view?usp=drivesdk
Ex. 6.6 Differentiation and integration of transforms . https://drive.google.com/file/d/108_Q_Wop_M5vTEWxnmgYJyCl07n47F_O/view?usp=drivesdk

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