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Current time:0:00Total duration:6:31

Worked example: linear solution to differential equation

Video transcript

so let's get a little bit more comfort in our understanding of what a differential equation even is and so here we have a differential equation and we haven't started exploring how we find the solutions for differential equations yet but let's just say you saw this and someone just walked up to you on the street and says hey I will give you a clue that there's a solution to this differential equation that is essentially a linear function where Y is equal to MX plus B and you just need to figure out the MS and the B's or maybe the M and the B that makes this makes this linear function satisfy this differential equation and what I now encourage you to do is pause the video and see if you can do it so I'm assuming you have paused it and had a go at it so let's think this through together so if we know that this kind of a solution can be described in this way we have to figure out some M's and B's here this is telling us that if we were to take the derivative of this with respect to X if we take the derivative of MX plus B with respect to X that that should be equal to negative 2 times X plus 3 times y well we know Y is this thing minus 5 and that should be true for all X's in order for this to be a solution to this differential equation remember a solution to a differential equation is not a value or a set of values it is a function or a set of functions so in order for this to be to satisfy this differential equation needs to be true for all of these X's here so let's work through it let's figure out first what are dy/dx is so dy/dx we'll just take the derivative here with respect to X dy/dx is derivative of M X with respect to X is just going to be M and of course derivative of B with respect to X is just a constant so it's just going to be 0 so dy/dx is M so we could write m is equal to negative 2x is equal to negative 2x plus 3 times instead of putting Y there I could write MX plus B remember Y is equal to MX plus B and this is a repeated reminder this has to be true for all X's MX plus B and then of course we have the -5 and so if you weren't able to solve it the first time I encourage you to start from here and now figure out what M with M and B needs to be in order for this equation right over here in order for this to be true for all X's in order for this to be true for all X's so assuming you have paused again and had a go at it so let's just just keep algebraically manipulating this and I'll just switch to one color here so we have M M is equal to negative 2x plus if we distribute this 3 we're going to have 3 M X plus 3b and then of course we're going to have minus 5 and now we can group the X terms so if we were to group if we were to group let me find a new color here maybe this blue so if we were to take these two and add them together that's going to be negative 2 plus 3 M times X or we could write this as 3 M minus 2 times X and then you have your constant terms so you have these terms right over here so plus 3b minus 5 and of course that's all going to be equal to M that's going to be equal to M now remember this needs to be true this needs to be true for all X's so notice over here I have some I have some coefficient times X on the right hand side but on the left hand side I have no X's so somehow this thing must disappear this this is a constant so it's completely reasonable it's completely reasonable that this constant could be equal to M but the only way that I get these X's to disappear so all I'm left with is an M is if this thing is equal to 0 let me say that again because I think it might maybe be a little bit counterintuitive what I'm about to do we're saying that M some constant value is equal to some coefficient times X plus some other constant value well in order for constant value to be equal to a coefficient times X plus some other constant value the coefficient on X must be equal to 0 another way to think about it is this should be you can rewrite the left-hand side here at 0x plus M and so you see you kind of match the coefficients so 0 must be equal to 3 M minus 2 and M is equal to 3 B M is equal to 3 B minus 5 M is equal to 3 B minus 5 so let's let's use that knowledge that information to solve for M and B so we could use this first one so 3 M minus 2 must be equal to 0 so let's write that 3 m minus 2 is equal to 0 or 3 M is equal to 2 or M is equal to 2/3 so we figure it out what M is and then we could use that information because we know that we know that M is equal to is equal to 3b minus 5 M is 2/3 so we get 2/3 is equal to 3 B minus 5 we could add 5 to both sides which is the same thing as adding 15 thirds to both sides is that did I do that right yeah it's adding 5 to both sides the same thing as adding 15 thirds to both sides so let's do that 15 over 3 plus 15 over 3 these cancel out that's just 5 right over there on the left-hand side we have 17 over 3 is equal to 3b or if you divide both sides by 3 you get B is equal to 17 B is equal to 17 over 9 and we're done we just found the a particular solution for this differential equation the solution is y is equal to 2/3 X plus 17 over 9 and I encourage you after watching this video to verify that this particular solution indeed does satisfy this differential equation for all X's for all X's