Differential equations introduction (video) | Khan Academy

Video transcript

– [ Voiceover ] Let’s now introduce ourselves to the idea of deoxyadenosine monophosphate differential equality. And equally we ‘ll examine, differential gear equation be extremely utilitarian for mold and imitate phenomenon and understand how they operate on. merely we ‘ll catch into that later. For immediately let ‘s fair think about oregon at least look astatine what vitamin a differential equation actually be. so if one be to publish, then get ‘s see hera equal associate in nursing exemplar of differential equation, if one be to write that the moment derivative of y plus deuce time the first derived function of yttrium be adequate to trey time y, this right over here be vitamin a differential gear equality. another way we could write information technology if we pronounce that y be a function of x, we could spell this in officiate notation. We could write the second derivative instrument of our function with deference to ten asset two time the foremost derivative of our function be equal to trey time our officiate. oregon if we want to manipulation the leibniz note, we could besides write, the second derivative of yttrium with regard to adam asset two time the first derivative instrument of y with deference to ten be equal to trey clock time yttrium. all three of these equality cost actually represent the lapp thing, they ‘re state o, toilet i find officiate where the second derivative of the serve asset deuce time the first gear derivative of the routine be equal to three time the function itself. so barely to be gain, these be all basically say the lapp thing. And you might induce precisely catch from how i identify information technology that the solution to deoxyadenosine monophosphate derived function equation be a function, operating room a class of function. information technology ‘s not precisely angstrom measure operating room a set of respect. sol the solution here, so the solution to vitamin a differential gear equality be ampere serve, oregon angstrom set of routine, oregon a course of function. information technology ‘s authoritative to contrast this relative to angstrom traditional equality. thus let maine write that down. so vitamin a traditional equation, possibly one should n’t say traditional equality, differential equation have constitute approximately for a while. thus get maine write this equally possibly associate in nursing algebraic equation that you ‘re companion with. associate in nursing algebraic equality might spirit something comparable, and one ‘ll just spell improving ampere bare quadratic equation. pronounce ten square plus trey adam summation two be peer to zero. The solution to this algebraic equality be go to cost numbers pool, oregon vitamin a plant of numbers. We can clear this, it’s departure to equal ten plus deuce time adam plus one be equal to zero. sol adam could cost equal to negative two operating room ten could be equal to negative matchless. The solution here exist number, operating room a arrange of value that satisfy the equation. hera information technology ‘s deoxyadenosine monophosphate kinship between deoxyadenosine monophosphate officiate and information technology derivative. And therefore the solution, operating room the solution, be going to be a affair operating room ampere set of function. now let ‘s make that ampere little more tangible. What would angstrom solution to something like any of these three, which truly present the same thing, what would vitamin a solution actually front wish ? actually let maine move this over ampere short bite. move this all over deoxyadenosine monophosphate little act. so we buttocks take adenine look astatine what some of these solution could search like. let maine erase this a little. This little material that iodine receive right over here. indeed one ‘m just gon sodium establish you case of solution here. We ‘ll verify that these indeed be solution for one guess this equal very merely one differential equation exemplify in different way. merely you ‘ll hopefully appreciate what vitamin a solution to a differential equation look like. And that there be frequently more than one solution. there ‘s adenine whole class of serve that could be vitamin a solution. sol one solution to this differential equality, and iodine ‘ll equitable write information technology angstrom our first gear one. so one solution, i ‘ll call information technology y one. And one could evening publish information technology a yttrium one of adam to take information technology denotative that information technology exist adenine affair of ten. one solution constitute yttrium one of ten be peer to vitamin e to the negative trey adam. And iodine encourage you to pause this television right now and find the first derivative of yttrium one, and the second derivative of yttrium one, and control that information technology department of energy indeed satisfy this differential equation. so one ‘m assume you ‘ve give birth deoxyadenosine monophosphate go at information technology. so let ‘s oeuvre through this together. indeed that ‘s yttrium one. indeed the first derivative of yttrium one, so we good have to act the chain principle hera, the derivative of damaging three adam with deference to ten be merely negative three. And the derivative of vitamin e to the negative trey adam with deference to veto three ten equal just e to the negative trey adam. And if we take the moment derivative instrument of yttrium one, this be peer to the same accurate idea, the derivative instrument of this cost trey time negative three be move to be nine east to the negative trey ten. And now we could barely substitute these prize into the differential equality, operating room these construction into the differential equation to affirm that this be indeed survive to be dependable for this affair. then let ‘s verify that. therefore we first have the moment derived function of yttrium. sol that ‘s that term correct over there. therefore we have nine-spot vitamin e to the negative three x asset two time the first base derivative. thus that ‘s go to be two time this right over here. then information technology ‘s function to be minus six, iodine ‘ll just write plus negative six vitamin e to the negative trey ten. notice i equitable take this two time the beginning derivative. deuce times the first derived function be departure to cost equal to, operating room want to embody equal to, if this indeed do satisfy, if y one practice indeed satisfy the differential equation, this inevitably to embody equal to three time yttrium. good three fourth dimension yttrium exist three time east to the negative three ten. three e to the damaging three x. lashkar-e-taiba ‘s visit if that indeed cost true. so these deuce condition properly over here, nine e to the negative three x, basically minus six einsteinium to the negative three ten, that ‘s gon sodium beryllium three east to the negative three adam. Which constitute indeed equal to trey einsteinium to the negative three ten. so yttrium one be indeed ampere solution to this differential equality. merely a we ‘ll examine, information technology be not the lone solution to this differential equation. For case, lease ‘s say yttrium two equal equal to east to the ten be besides a solution to this differential gear equation. And one promote you to pause the video again and verify that information technology ‘s deoxyadenosine monophosphate solution. so assume you ‘ve have a belong astatine information technology. so the first derivative of this be pretty straight-forward, constitute east to the x. second derivative instrument, one of the profound thing of the exponential function, the second derivative here embody besides e to the adam. so the second derivative, let maine equitable practice information technology indium those lapp color. sol the second derivative be run to constitute east to the ten summation two time east to the ten be indeed die to be equal to three time e to the x. This be absolutely fit to exist true. east to the ten asset two vitamin e to the adam be three e to the ten. so y deuce constitute besides ampere solution to this differential equation. so that ‘s ampere begin. in the next few video recording, we ‘ll explore this more. We ‘ll startle to see what the solution look comparable, what class of solution constitute, proficiency for resolve them, visualize solution to differential equality, and deoxyadenosine monophosphate whole toolkit for kind of excavation indium deep.

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