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Q-5) [10 Marks] Solve the following non-homogenous differential equation using Laplace Transform. +y = cost (hint: at one step you will need t0 use convolution)_...

Question

Q-5) [10 Marks] Solve the following non-homogenous differential equation using Laplace Transform. +y = cost (hint: at one step you will need t0 use convolution)_

Q-5) [10 Marks] Solve the following non-homogenous differential equation using Laplace Transform. +y = cost (hint: at one step you will need t0 use convolution)_



Answers

In Exercises $1-6,$ use the results of this section to calculate the specified Laplace transforms and state the corresponding inverse transforms. What restrictions on the transform variable $s$ apply? Assume $a$ and $b$ are real constants, $n$ is a nonnegative integer, and $i$ is the imaginary unit $\left(i^{2}=-1\right)$ $$L\left\{e^{a t} \cos (b t)\right\}$$

Hello. You have to find a lot of this given function. We know that some of cause me it is as a part as a square place Biscoe and we know that The lab plus through that is elaboration from Italy power 80 I hope the So this will be cost too. FF has -8. So we will use this system here. We love to help you to the power Beauty because Bt So this will be cost as minus a divide by has minus a squared plus B squared. Where has it greater than a. This is the restriction. I hope you understood. Thank you.

Let's use convolution formula. So what is the convolution formula? F. Star G of T is negative infinity to infinity. The dow The T -4 detail. So this is the form of a convolution too. Time domain signals. This is one very important property in laplace transform transform of the convolution. Two signals is the product of the individual that was transports. Now it's going back the last transform its offices one by as into SPS too. Let's choose to focus as one by space too. So what is it in muslim pass transform? It is for you is or have a real like Applebee's bar minus to B. You are the your step function U. Is a unit sub function. And whereas what is the Gfs? Gfs is one by us. It was laplace transform is um is or you often. All right so now let's use this popular to them. So we got start of G. F. D. His integration of minds infinity to infinity. How to will be power -2 Tao. You have to and G of Travis GFT minus two hours. You have t minus style data but your t minus top is Equal one when T- Star was positive or how is less in T. Whereas this is one when you have to use one when is greater than zero so how greater than zero towel? Less than the only when tom is greater than zero is less than the the product is one. So that means integration of limits will be zero to T Iban -2. Top. And this is why this is one data. Now let's integrate. So what is integration of the par minus two? It's a par minus 2001 minus two. Then just substitute upper and lower limits. It's the par minus two by minus two minus of Subject. Our zero Evans is always one x -2 so it's nothing but half of one minus the par minus two. So this is the final answer. Who's I mean, it was like last transformer off one by is into SPS to buy convolution kim.


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