ESE 318-02, Fall 2014Lecture 4, Sept. 4, 2014Convolution, Integro-Differential Equations, Systems of ODEsReview. We recently derived these properties and transforms.     sFeatatfas U      atgeattgasLU   100tettst         sFdsdtftsFdsdttfnnnn1        tfTtfifdttfeetfTstsT011L tf sF ttf sFdsdSomewhat tricky problem: Find f(t), the inverse Laplace transform of F (s), where  419ln22sssF               ttettfttesFdsdttftetsFdsdsssssssssFdsdsssFSolutionttt3cos2cos23cos22cos22cos23cos241129241129241ln9ln:11222222LLESE 318-02, Fall 2014Another one: Find f(t), the inverse Laplace transform of F(s), where sbasbasF arctanarctan2121without using Line 45 of the Table in Zill!               tbtattfbtvatubtatsFdsdttfvuvuvuTrigtbatbasFdsdbasbabasbasbasbasbasbasFdsdSolutioncossincossin)sin()sin(21cossin:sinsin212111211121:121122222222LLIf you see a Laplace transform involving ln or arctan, think of taking the derivative.Convolution. The convolution of two functions f(t) and g(t) is denoted as f*g and is defined below. Note that f and g are both functions of time, and so is the convolution.       tdtgftgtf0A property of the convolution is that it is commutative, meaning we can switch how we handle f and g inside the integral.   tdgtffggf0I’ll derive the Convolution Theorem since the proof in Zill is full of typos.ESE 318-02, Fall 2014                      gfdtdtgfedtdtgfeinnerintddttgfeddgfedegdefsGsFgftsttststsss   L0 00 000 000int  That 4th line was due to changing the order of integration when integrating over the region in the t- plane. (We’ll cover such integration later in the semester, in case your Calculus is rusty.)To summarize:               sFsGsGsFtgtfgf  LLLLook at the special case where g(t) = 1.          ssFfdfdfdtgtfgfttt11000LLThis makes sense since taking the derivative meant multiplying by s in the Laplace domain; taking the integral means dividing by s. (Note here that the integral is a definite integral that starts at time t = 0 and is 0 for t < 0.)We can use the above properties to evaluate convolutions without taking integrals.ESE 318-02, Fall 2014Zill 4.4.28. (Use lines 7 and 8 of the table. I will go further and find the integral itself by using line 22 of the Table for the inverse transform.)         ttsssssssttttdttsin12211111cossincossincossin212222220LLLLZill 4.4.26. (First use Table, line 22. I will go further, and evaluate the integral itself by taking the inverse Laplace transform, using The Table, line 25.)    tttdssssttsdttcossinsin12121sin1sin022220LLWe can also solve equations involving integrals, convolutions or combinations of differentials, integrals and convolutions.Zill 4.4.44. Solve the given equation, which involves a convolution.         3121214242422222222000!31211212121211221211211111212tttfssssssFssFsFssFsssFsFsFFsefefdtfedtfedtfeetfttttttZill 4.4.41 Example of integral equation (or convolution with g(t)= 1).    ttetfsFsFssFssFsFdftf1111111110ESE 318-02, Fall 2014Zill 4.4.45 Example of integro-differential equation.            ttttysssssssssYssssYssssYssYssssYydyttytsinsin122111111111111111111100sin121222222222222220Systems of linear differential equations. Our method of using Laplace transforms extends pretty easily to systems of differential equations (in multiple variables) by transforming into corresponding multiple Laplace transforms. The algebra will be more difficult, generally, but the concepts are the same.Zill 4.6.3.     tttyttyyttttdtdxtttxsssssXssXssssXssssXsssXssXsXsYYXsYsYsXYXsXyxyxdtdyyxdtdx3sin3cos23sin3cos4223sin3cos3cos53sin33sin3cos933599515191511011141101112151212155211122120105237314353522222222222In the above solution, once we found x(t), we could plug it back into the original ODE and solve for y(t) in the time domain. Another way to do it is to solve for Y(s) in theESE 318-02, Fall 2014Laplace domain, just like we did for x, and then take the inverse transform. This was not done in class, but here it is for reference, starting with the Laplace domain equations for X and Y found above. 