Original subtitles

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Let's talk about solutions to differential equations.

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So solutions to these there's all types of solutions to ease of harm to all types of terminology.

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Say you have something like Y equals three X squared plus C.

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So this is called a one parameter

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one parameter family of solutions family of solutions so you have infinitely many solutions one for

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each choice of C say you had something like this C one each of the two x plus C to each of the five

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X. This is this would be called a two parameter family of solutions.

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So two parameter family solutions and you have infinitely many solutions again and all you have to do

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is you can pick the seasons infinitely many choices for the C's if you pick the C's say I pick C one

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equal to one that would give me either the two X and then C two equal to three that would give me each

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of the five X. This one doesn't have any constants and it doesn't have any arbitrary constants This

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is called a particular solution particular solution

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and it's free from arbitrary parameters that's why it's called the particular solution it has has no

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CS so free from arbitrary

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parameters.

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So if it has one C it's called a one parameter of family of solutions if it has two seas it's called

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a two parameter family of solutions if it doesn't have any CS It's called a particular solution and

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there's one more that's really important y equals zero.

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This is a special case whenever it's zero we call it the trivial solution so the trivial solution

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all of these are called explicit solutions explicit solutions because we have Y explicitly defined in

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terms of X. So.

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Right.

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They all look like Y equal stuff with X is only right.

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So we have specifically solved for wives.

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These are all explicit solutions.

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If Y is not explicitly given by a formula in terms of X or the other independent variable we'll just

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use X for clarity.

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You have what's called an implicit solution for example.

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We had Alan X plus Y minus Y squared equals each of the X plus X plus C.

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This here is not an explicit solution.

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This is called an implicit solution.

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So this is an implicit solution.

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OK so explicit means you you solve for y.

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Let's do a simple example to illustrate another key concept

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say we have this differential equation x y prime plus y equals zero and we want to show that Y equals

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one over X is a solution

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to this dy and we're also gonna find what's called the interval of definitions find the interval of

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definition

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okay.

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So to show it's a solution all we have to do is plug this function into this differential equation.

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So solution step are writing down the function again.

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So why is equal to 1 over x.

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And now we'll take the derivative of this to do that.

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You can write it as X to the negative one right you can take this and bring it upstairs and take the

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derivative you would use the power rule so you put the negative one in the front.

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If you get negative x the negative 1 minus 1 is negative 2.

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So you get negative 1 over X squared so y prime is equal to negative 1 over X squared.

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So now all we have to do is plug it into our differential equation.

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So we have X Y prime plus y sets equal to x and y prime we said was this right here.

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So this is negative 1 over X squared plus Y and then Y is right here.

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So 1 over X OK so all we've done is plug in the function and its derivative into our D axis cancels.

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We get negative one over x plus one over X and it's equal to zero.

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Hurrah.

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So it checks.

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We started with this and we showed it's equal to zero which is the deal here.

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So it is indeed a solution.

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That's what it means for a function to be a solution.

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OK.

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Now we have to find the interval of definition.

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So to do that I'm going to graph one over X so one over X looks like this.

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And it has a vertical asymptote at zero and a horizontal asymptote at zero.

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And it looks like this.

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And so we have to find what's called the interval of definition.

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Now the domain of this function is negative infinity to zero union zero to infinity.

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All right.

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So it's everything except zero.

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But that is not the interval of definition the interval of definition is the largest interval over which

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the solution is defined.

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So this here is not an interval OK.

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By definition interval is a set where if you pick any two numbers in that set every number between them

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is also in that set this set here does not satisfy that.

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Right.

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Because I can pick negative 1 and 2 and then zero fails.

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Right zero is not in the set.

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So this is not an interval.

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So what is the interval of definition.

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Well it's up to us.

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Actually we can pick this one so we can pick negative infinity to zero or we can pick this one so we

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can pick zero to infinity.

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So I'm just going to pick zero to infinity and we'll call that the interval of definition.

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So you get to pick whichever one you like most of the time.

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So I hope that made sense.

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Let's let's do one more example let's do one more example since we're talking about solutions.

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Let's finish talking about solutions there's one other key idea.

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Say we have this this differential equation d y the X equals X square root of Y.

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And what we know that a one parameter family of solutions

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family of solutions

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is given by this Y equals 1 fourth X squared plus C quantity squared.

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So this is a nonlinear D because we have that square root of Y.

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So why did the 1 1/2 power it's actually really easy to solve.

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And they're giving us the answer and the question is is there another

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solution

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so is there another solution to this differential equation.

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Right.

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That's not given by this one parameter family.

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So how would you figure that out.

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Well just observation the answer is yes and the answer is y equals zero Y equals zero is a solution

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to this differential equation.

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You can check if Y equals zero then d y the X while the derivative is also zero because it's a constant

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write the derivative of zero zero.

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So if you take Y equals zero and take d y the x equals zero and plug them back in here you get zero

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equals x times the square root of zero.

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So you get zero equals zero.

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Yep.

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So zero is a solution.

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So this is the solution that we can't get by picking C..

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All right.

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We can't get Y equals.

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There's no there's no c we can pick that's going to make this zero.

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So this is called a singular solution.

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So whenever you have a solution that you can't get by picking values of C it's called a singular solution.

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So we can't get it.

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We cannot get this

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by picking C. It's impossible.

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So whenever you have a solution that you cannot get by picking values of C it is called a singular solution

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super super important.

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If it turns out to be the case that you can get all the solutions by picking values of C you have what's

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called the general solution let me write that down.

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It's really important.

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So if we can get all solutions

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that you can get every single solution by picking C we have what's called the general solution.

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We have the general solution.

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So in this case this one parameter family here is not the general solution.

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Right.

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Because here is a solution that we could not get by picking values of seat.

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Hence the name singular.

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So it's a singular solution.

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All right.

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If it were the case that we can get all the solutions then we would have what was called the general

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solution.

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Whenever you have a linear differential equation you always have the general solution that's a big big

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result from differential equations.

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I hope that video made sense with a lot of information and in the next video we'll briefly talk about

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what's called initial Valley problems.

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That's it.