align:start position:0% hi everyone welcome back so today I'd align:start position:0% hi everyone welcome back so today I'd align:start position:0% hi everyone welcome back so today I'd like to take a look at solving a linear align:start position:0% like to take a look at solving a linear align:start position:0% like to take a look at solving a linear o de but with a periodic input and align:start position:0% o de but with a periodic input and align:start position:0% o de but with a periodic input and specifically we're asked to find one align:start position:0% specifically we're asked to find one align:start position:0% specifically we're asked to find one solution that a particular solution align:start position:0% solution that a particular solution align:start position:0% solution that a particular solution which is also the periodic solution of align:start position:0% which is also the periodic solution of align:start position:0% which is also the periodic solution of the differential equation X dot plus two align:start position:0% the differential equation X dot plus two align:start position:0% the differential equation X dot plus two x dot plus 4x equals align:start position:0% x dot plus 4x equals align:start position:0% x dot plus 4x equals the square wave function so the square align:start position:0% the square wave function so the square align:start position:0% the square wave function so the square wave function is periodic function with align:start position:0% wave function is periodic function with align:start position:0% wave function is periodic function with period 2pi it's defined as minus 1 and 1 align:start position:0% period 2pi it's defined as minus 1 and 1 align:start position:0% period 2pi it's defined as minus 1 and 1 on the intervals minus prior to 0 and 0 align:start position:0% on the intervals minus prior to 0 and 0 align:start position:0% on the intervals minus prior to 0 and 0 to PI and we know that the square wave align:start position:0% to PI and we know that the square wave align:start position:0% to PI and we know that the square wave function has the following Fourier align:start position:0% function has the following Fourier align:start position:0% function has the following Fourier series align:start position:0% series align:start position:0% series so let you think about this problem for align:start position:0% so let you think about this problem for align:start position:0% so let you think about this problem for a moment and I'll come back in a second align:start position:0% align:start position:0% hi everyone welcome back align:start position:0% hi everyone welcome back align:start position:0% hi everyone welcome back okay so the reason we've been studying align:start position:0% okay so the reason we've been studying align:start position:0% okay so the reason we've been studying Fourier series is to essentially solve align:start position:0% Fourier series is to essentially solve align:start position:0% Fourier series is to essentially solve differential equations with complicated align:start position:0% differential equations with complicated align:start position:0% differential equations with complicated forcing inputs on the right hand side align:start position:0% forcing inputs on the right hand side align:start position:0% forcing inputs on the right hand side which are periodic and the reason we've align:start position:0% which are periodic and the reason we've align:start position:0% which are periodic and the reason we've been studying Fourier series is because align:start position:0% been studying Fourier series is because align:start position:0% been studying Fourier series is because we know that differential equations with align:start position:0% we know that differential equations with align:start position:0% we know that differential equations with sines and cosines as forcing terms on align:start position:0% sines and cosines as forcing terms on align:start position:0% sines and cosines as forcing terms on the right-hand side are relatively easy align:start position:0% the right-hand side are relatively easy align:start position:0% the right-hand side are relatively easy to solve and align:start position:0% to solve and align:start position:0% to solve and we want to be able to solve the same align:start position:0% we want to be able to solve the same align:start position:0% we want to be able to solve the same differential equation with a more align:start position:0% differential equation with a more align:start position:0% differential equation with a more complicated periodic function on the align:start position:0% complicated periodic function on the align:start position:0% complicated periodic function on the right hand side so the general approach align:start position:0% right hand side so the general approach align:start position:0% right hand side so the general approach is to first align:start position:0% align:start position:0% do you compose align:start position:0% align:start position:0% the right-hand side align:start position:0% align:start position:0% into a Fourier series align:start position:0% align:start position:0% okay and this step is essentially align:start position:0% okay and this step is essentially align:start position:0% okay and this step is essentially already done for us we're told what the align:start position:0% already done for us we're told what the align:start position:0% already done for us we're told what the Fourier series is align:start position:0% Fourier series is align:start position:0% Fourier series is and what we do is we solve align:start position:0% and what we do is we solve align:start position:0% and what we do is we solve the Eau de align:start position:0% the Eau de align:start position:0% the Eau de X dot dot plus 2 X dot plus 4x and align:start position:0% X dot dot plus 2 X dot plus 4x and align:start position:0% X dot dot plus 2 X dot plus 4x and I'm just going to take one align:start position:0% I'm just going to take one align:start position:0% I'm just going to take one term of the Fourier series sine NT align:start position:0% align:start position:0% so if we know the right hand side is a align:start position:0% so if we know the right hand side is a align:start position:0% so if we know the right hand side is a sum of a whole bunch of signs what we're align:start position:0% sum of a whole bunch of signs what we're align:start position:0% sum of a whole bunch of signs what we're going to do is we're going to solve for align:start position:0% going to do is we're going to solve for align:start position:0% going to do is we're going to solve for any specific one of those signs so we align:start position:0% any specific one of those signs so we align:start position:0% any specific one of those signs so we want to solve this differential equation align:start position:0% want to solve this differential equation align:start position:0% want to solve this differential equation and align:start position:0% and align:start position:0% and then we use align:start position:0% align:start position:0% superposition align:start position:0% align:start position:0% so if we know what the solution is to 1 align:start position:0% so if we know what the solution is to 1 align:start position:0% so if we know what the solution is to 1 sine NT and we know that the right hand align:start position:0% sine NT and we know that the right hand align:start position:0% sine NT and we know that the right hand side forcing is a sum of many sign align:start position:0% side forcing is a sum of many sign align:start position:0% side forcing is a sum of many sign entities with appropriate weight factors align:start position:0% entities with appropriate weight factors align:start position:0% entities with appropriate weight factors then we can use superposition to align:start position:0% then we can use superposition to align:start position:0% then we can use superposition to construct align:start position:0% align:start position:0% a align:start position:0% align:start position:0% final solution align:start position:0% align:start position:0% okay so this is the method to attack align:start position:0% okay so this is the method to attack align:start position:0% okay so this is the method to attack this problem align:start position:0% this problem align:start position:0% this problem alright so we've already done step 1 or align:start position:0% alright so we've already done step 1 or align:start position:0% alright so we've already done step 1 or we were given step 1 and we wanted to align:start position:0% we were given step 1 and we wanted to align:start position:0% we were given step 1 and we wanted to solve step 2 and we can solve this just align:start position:0% solve step 2 and we can solve this just align:start position:0% solve step 2 and we can solve this just using the exponential response formula align:start position:0% using the exponential response formula align:start position:0% using the exponential response formula so what we're going to do is we're just align:start position:0% so what we're going to do is we're just align:start position:0% so what we're going to do is we're just going to align:start position:0% going to align:start position:0% going to complexity the right hand side align:start position:0% align:start position:0% so if I want to solve the differential align:start position:0% so if I want to solve the differential align:start position:0% so if I want to solve the differential equation x + 2 X dot plus 4x equals sign align:start position:0% equation x + 2 X dot plus 4x equals sign align:start position:0% equation x + 2 X dot plus 4x equals sign in T I'm going to denote the solution align:start position:0% in T I'm going to denote the solution align:start position:0% in T I'm going to denote the solution with a subscript n and it's going to be align:start position:0% with a subscript n and it's going to be align:start position:0% with a subscript n and it's going to be the imaginary part of align:start position:0% the imaginary part of align:start position:0% the imaginary part of and I'm just using the exponential align:start position:0% and I'm just using the exponential align:start position:0% and I'm just using the exponential response formula 1 over the align:start position:0% response formula 1 over the align:start position:0% response formula 1 over the characteristic polynomial evaluated at I align:start position:0% align:start position:0% ne2 the int align:start position:0% ne2 the int align:start position:0% ne2 the int you might ask how did I get that align:start position:0% align:start position:0% well I basically just took the align:start position:0% well I basically just took the align:start position:0% well I basically just took the complexity aight formula or sorry the align:start position:0% complexity aight formula or sorry the align:start position:0% complexity aight formula or sorry the complex if I'd equation align:start position:0% align:start position:0% and noted that sine NT was the imaginary align:start position:0% and noted that sine NT was the imaginary align:start position:0% and noted that sine NT was the imaginary part of e to the int and then use the align:start position:0% part of e to the int and then use the align:start position:0% part of e to the int and then use the exponential response formula and I want align:start position:0% exponential response formula and I want align:start position:0% exponential response formula and I want to take the imaginary part at the end of align:start position:0% to take the imaginary part at the end of align:start position:0% to take the imaginary part at the end of the day align:start position:0% align:start position:0% so what is a the characteristic align:start position:0% so what is a the characteristic align:start position:0% so what is a the characteristic polynomial in this case it's s squared align:start position:0% polynomial in this case it's s squared align:start position:0% polynomial in this case it's s squared plus 2 s plus 4 align:start position:0% plus 2 s plus 4 align:start position:0% plus 2 s plus 4 which means that P of I n align:start position:0% which means that P of I n align:start position:0% which means that P of I n is going to be negative n squared plus 4 align:start position:0% is going to be negative n squared plus 4 align:start position:0% is going to be negative n squared plus 4 so I'll just group the real terms align:start position:0% so I'll just group the real terms align:start position:0% so I'll just group the real terms together plus 2i n align:start position:0% together plus 2i n align:start position:0% together plus 2i n so the 2i n comes from the 2's term align:start position:0% so the 2i n comes from the 2's term align:start position:0% so the 2i n comes from the 2's term and align:start position:0% and align:start position:0% and then X of n is going to be the imaginary align:start position:0% then X of n is going to be the imaginary align:start position:0% then X of n is going to be the imaginary part of align:start position:0% align:start position:0% 1 over 4 minus n squared align:start position:0% 1 over 4 minus n squared align:start position:0% 1 over 4 minus n squared plus 2i align:start position:0% plus 2i align:start position:0% plus 2i ne2 the int align:start position:0% ne2 the int align:start position:0% ne2 the int and align:start position:0% and align:start position:0% and I'm going to use the amplify align:start position:0% I'm going to use the amplify align:start position:0% I'm going to use the amplify the amplitude phase form to convert this align:start position:0% the amplitude phase form to convert this align:start position:0% the amplitude phase form to convert this or to convert this Cartesian complex align:start position:0% or to convert this Cartesian complex align:start position:0% or to convert this Cartesian complex number align:start position:0% number align:start position:0% number sorry I'm going to convert this align:start position:0% sorry I'm going to convert this align:start position:0% sorry I'm going to convert this Cartesian complex number into an align:start position:0% Cartesian complex number into an align:start position:0% Cartesian complex number into an amplitude phase form just because it's align:start position:0% amplitude phase form just because it's align:start position:0% amplitude phase form just because it's going to make taking the imaginary part align:start position:0% going to make taking the imaginary part align:start position:0% going to make taking the imaginary part of the solution very easy at the end of align:start position:0% of the solution very easy at the end of align:start position:0% of the solution very easy at the end of the day align:start position:0% align:start position:0% so the amplitude of this complex number align:start position:0% so the amplitude of this complex number align:start position:0% so the amplitude of this complex number is for n sorry 4 minus N squared align:start position:0% is for n sorry 4 minus N squared align:start position:0% is for n sorry 4 minus N squared quantity squared align:start position:0% quantity squared align:start position:0% quantity squared plus the imaginary part squared so align:start position:0% plus the imaginary part squared so align:start position:0% plus the imaginary part squared so that's 4n squared align:start position:0% that's 4n squared align:start position:0% that's 4n squared square rooted and align:start position:0% square rooted and align:start position:0% square rooted and then we have e align:start position:0% then we have e align:start position:0% then we have e upstairs e align:start position:0% upstairs e align:start position:0% upstairs e int and downstairs is going to be e I n align:start position:0% int and downstairs is going to be e I n align:start position:0% int and downstairs is going to be e I n Phi sorry i phi and align:start position:0% Phi sorry i phi and align:start position:0% Phi sorry i phi and i'm going to align:start position:0% i'm going to align:start position:0% i'm going to put a subscript n on the phi just align:start position:0% put a subscript n on the phi just align:start position:0% put a subscript n on the phi just because for each align:start position:0% because for each align:start position:0% because for each for each complex number we're going to align:start position:0% for each complex number we're going to align:start position:0% for each complex number we're going to have a different phase Phi and that align:start position:0% have a different phase Phi and that align:start position:0% have a different phase Phi and that phase Phi n is align:start position:0% phase Phi n is align:start position:0% phase Phi n is going to be the arctangent of align:start position:0% going to be the arctangent of align:start position:0% going to be the arctangent of 2 n align:start position:0% 2 n align:start position:0% 2 n divided divided by 4 minus n squared align:start position:0% align:start position:0% like that align:start position:0% align:start position:0% okay so now i can combine phi then with align:start position:0% okay so now i can combine phi then with align:start position:0% okay so now i can combine phi then with the upstairs term int in the exponent align:start position:0% the upstairs term int in the exponent align:start position:0% the upstairs term int in the exponent and when I take the imaginary part I'm align:start position:0% and when I take the imaginary part I'm align:start position:0% and when I take the imaginary part I'm only going to be left with sine of n t align:start position:0% only going to be left with sine of n t align:start position:0% only going to be left with sine of n t minus Phi n align:start position:0% minus Phi n align:start position:0% minus Phi n so X event align:start position:0% so X event align:start position:0% so X event yep going is going to be 1 over align:start position:0% yep going is going to be 1 over align:start position:0% yep going is going to be 1 over 4 minus N squared align:start position:0% 4 minus N squared align:start position:0% 4 minus N squared squared plus 4n squared align:start position:0% align:start position:0% quantity squared square rooted align:start position:0% quantity squared square rooted align:start position:0% quantity squared square rooted times sine of n t minus Phi n align:start position:0% times sine of n t minus Phi n align:start position:0% times sine of n t minus Phi n where Phi n was given using the align:start position:0% where Phi n was given using the align:start position:0% where Phi n was given using the arctangent formula and align:start position:0% arctangent formula and align:start position:0% arctangent formula and this gives us a solution which note is align:start position:0% this gives us a solution which note is align:start position:0% this gives us a solution which note is periodic with period 2 t.o align:start position:0% periodic with period 2 t.o align:start position:0% periodic with period 2 t.o right I i should also note that 5n is align:start position:0% right I i should also note that 5n is align:start position:0% right I i should also note that 5n is between align:start position:0% align:start position:0% 0 and 2 pi align:start position:0% 0 and 2 pi align:start position:0% 0 and 2 pi sorry 0 it and pie align:start position:0% sorry 0 it and pie align:start position:0% sorry 0 it and pie and align:start position:0% and align:start position:0% and this gives us this one solution to the align:start position:0% this gives us this one solution to the align:start position:0% this gives us this one solution to the differential equation with a forcing of align:start position:0% differential equation with a forcing of align:start position:0% differential equation with a forcing of sine NT on the right hand side align:start position:0% sine NT on the right hand side align:start position:0% sine NT on the right hand side so now what we want to do is we want to align:start position:0% so now what we want to do is we want to align:start position:0% so now what we want to do is we want to sum up many of these solutions using the align:start position:0% sum up many of these solutions using the align:start position:0% sum up many of these solutions using the superposition principle so if I go back align:start position:0% superposition principle so if I go back align:start position:0% superposition principle so if I go back now align:start position:0% now align:start position:0% now i'm going to write the original align:start position:0% i'm going to write the original align:start position:0% i'm going to write the original differential equation align:start position:0% align:start position:0% so i'm just going to go back and just align:start position:0% so i'm just going to go back and just align:start position:0% so i'm just going to go back and just rewrite this align:start position:0% rewrite this align:start position:0% rewrite this and i'm going to write the right-hand align:start position:0% and i'm going to write the right-hand align:start position:0% and i'm going to write the right-hand side using its fourier series 1 over n align:start position:0% side using its fourier series 1 over n align:start position:0% side using its fourier series 1 over n sine NT align:start position:0% sine NT align:start position:0% sine NT where n is odd align:start position:0% where n is odd align:start position:0% where n is odd ok and this is the problem we originally align:start position:0% ok and this is the problem we originally align:start position:0% ok and this is the problem we originally wanted to solve and essentially what align:start position:0% wanted to solve and essentially what align:start position:0% wanted to solve and essentially what we've done is we've solved the problem align:start position:0% we've done is we've solved the problem align:start position:0% we've done is we've solved the problem for each individual sine NT so how do we align:start position:0% for each individual sine NT so how do we align:start position:0% for each individual sine NT so how do we get the full solution well what we have align:start position:0% get the full solution well what we have align:start position:0% get the full solution well what we have to do is we have to multiply the align:start position:0% to do is we have to multiply the align:start position:0% to do is we have to multiply the solution for each sign in T by a factor align:start position:0% solution for each sign in T by a factor align:start position:0% solution for each sign in T by a factor of 4 divided by PI and 1 over N and then align:start position:0% of 4 divided by PI and 1 over N and then align:start position:0% of 4 divided by PI and 1 over N and then we have to add all of these solutions up align:start position:0% we have to add all of these solutions up align:start position:0% we have to add all of these solutions up for all odd values of n align:start position:0% align:start position:0% so for example X is just going to be align:start position:0% so for example X is just going to be align:start position:0% so for example X is just going to be four over pi align:start position:0% four over pi align:start position:0% four over pi sum of n odd align:start position:0% sum of n odd align:start position:0% sum of n odd one over n times the solution to every align:start position:0% one over n times the solution to every align:start position:0% one over n times the solution to every sine NT align:start position:0% sine NT align:start position:0% sine NT which we've already computed and I've align:start position:0% which we've already computed and I've align:start position:0% which we've already computed and I've denoted as X sub N and align:start position:0% denoted as X sub N and align:start position:0% denoted as X sub N and X sub n is up here align:start position:0% X sub n is up here align:start position:0% X sub n is up here so if we want to be explicit about it align:start position:0% so if we want to be explicit about it align:start position:0% so if we want to be explicit about it I'll write the whole thing out as align:start position:0% I'll write the whole thing out as align:start position:0% I'll write the whole thing out as for over PI and odd align:start position:0% align:start position:0% one over align:start position:0% one over align:start position:0% one over four minus N squared quantity squared align:start position:0% four minus N squared quantity squared align:start position:0% four minus N squared quantity squared plus 4n squared align:start position:0% plus 4n squared align:start position:0% plus 4n squared square rooted align:start position:0% square rooted align:start position:0% square rooted we also have a factor of n out front align:start position:0% we also have a factor of n out front align:start position:0% we also have a factor of n out front and we have a sine NT align:start position:0% and we have a sine NT align:start position:0% and we have a sine NT minus Phi n align:start position:0% align:start position:0% okay so there's the final answer for a align:start position:0% okay so there's the final answer for a align:start position:0% okay so there's the final answer for a particular solution in its full glorious align:start position:0% particular solution in its full glorious align:start position:0% particular solution in its full glorious detail we can also check that this align:start position:0% detail we can also check that this align:start position:0% detail we can also check that this particular solution is periodic note how align:start position:0% particular solution is periodic note how align:start position:0% particular solution is periodic note how each sign in T is periodic and each sine align:start position:0% each sign in T is periodic and each sine align:start position:0% each sign in T is periodic and each sine NT has period of at least two pi so when align:start position:0% NT has period of at least two pi so when align:start position:0% NT has period of at least two pi so when we sum up a whole bunch of functions align:start position:0% we sum up a whole bunch of functions align:start position:0% we sum up a whole bunch of functions which all have a period of at least two align:start position:0% which all have a period of at least two align:start position:0% which all have a period of at least two pi the the sum is also going to be a align:start position:0% pi the the sum is also going to be a align:start position:0% pi the the sum is also going to be a periodic function with at least two pi align:start position:0% periodic function with at least two pi align:start position:0% periodic function with at least two pi and align:start position:0% and align:start position:0% and and as a result this gives us that the align:start position:0% and as a result this gives us that the align:start position:0% and as a result this gives us that the the answer we're looking for also in align:start position:0% the answer we're looking for also in align:start position:0% the answer we're looking for also in addition if we want the full general align:start position:0% addition if we want the full general align:start position:0% addition if we want the full general solution to the differential equation to align:start position:0% solution to the differential equation to align:start position:0% solution to the differential equation to this particular solution we also have to align:start position:0% this particular solution we also have to align:start position:0% this particular solution we also have to add the homogeneous piece align:start position:0% add the homogeneous piece align:start position:0% add the homogeneous piece so this concludes the problem and I'll align:start position:0% so this concludes the problem and I'll align:start position:0% so this concludes the problem and I'll just quickly recap when solving a align:start position:0% just quickly recap when solving a align:start position:0% just quickly recap when solving a differential equation with the periodic align:start position:0% differential equation with the periodic align:start position:0% differential equation with the periodic forcing function on the right hand side align:start position:0% forcing function on the right hand side align:start position:0% forcing function on the right hand side again to iterate the steps you first align:start position:0% again to iterate the steps you first align:start position:0% again to iterate the steps you first before you decompose the right hand side align:start position:0% before you decompose the right hand side align:start position:0% before you decompose the right hand side into summation of sines and cosines you align:start position:0% into summation of sines and cosines you align:start position:0% into summation of sines and cosines you then solve the differential equation for align:start position:0% then solve the differential equation for align:start position:0% then solve the differential equation for sine NT cosine NT align:start position:0% sine NT cosine NT align:start position:0% sine NT cosine NT individually this gives you a solution align:start position:0% individually this gives you a solution align:start position:0% individually this gives you a solution for each term on the right hand side of align:start position:0% for each term on the right hand side of align:start position:0% for each term on the right hand side of the differential equation and then at align:start position:0% the differential equation and then at align:start position:0% the differential equation and then at the end of the day you superposition to align:start position:0% the end of the day you superposition to align:start position:0% the end of the day you superposition to sum up all this all of the solutions and align:start position:0% sum up all this all of the solutions and align:start position:0% sum up all this all of the solutions and that gives you one final big solution align:start position:0% that gives you one final big solution align:start position:0% that gives you one final big solution okay so i hope you enjoyed this problem align:start position:0% okay so i hope you enjoyed this problem align:start position:0% okay so i hope you enjoyed this problem and i'll see you next time