I've spoken a lot about second order linear homogeneous
differential equations in abstract terms, and how if g
is a solution, then some constant
times g is also a solution.
Or if g and h are solutions, then g
plus h is also a solution.
Let's actually do problems, because I think that will
actually help you learn, as opposed to
help you get confused.
So let's say I have this differential equation, the
second derivative of y, with respect to x, plus 5 times the
first derivative of y, with respect to x, plus 6 times y
is equal to 0.
So we need to find a y where 1 times its second derivative,
plus 5 times its first derivative, plus 6 times
itself, is equal to 0.
And now, let's just do a little bit of-- take a step
back and think about what kind of function-- if I have the
function and I take its derivative and then I take its
second derivative, most times I get
something completely different.
Like, if y was x squared, then y prime would be 2x, and y
prime prime would be 2.
And then to add them together you'd say, well, how would my
x terms cancel out so that you get 0 in the end?
So draw back into your brain and think, is there some
function that when I take its first and second derivatives,
and third and fourth derivatives, it essentially
becomes the same function?
Maybe the constant in front of the function changes as I take
the derivative.
And if you've listened to a lot of my videos, you'd
realize that it probably is what I consider to be the most
amazing function in mathematics.
And that is the function e to the x.
And in particular, maybe e to the x won't work here-- or you
can even try it out, right?
If you did e to the x, it won't satisfy
this equation, right?
You would get e to the x, plus 5e to the x, plus 6e to the x.
That would not equal to 0.
But maybe y is equal to e to some constant r, times x.
Let's just make the assumption that y is equal to some
constant r times x, substitute it back into this, and then
see if we can actually solve for an r that makes this
equation true.
And if we can, we've found the solution, or maybe we've found
several solutions.
So let's try it out.
Let's try y is equal to e to the rx into this
differential equation.
So what is the first derivative of
it, first of all.
So y soon. prime is equal to what?
Derivative chain rule.
Derivative of the inside is r.
And then derivative of the outside is still
just e to the rx.
And what's the second derivative?
y prime prime is equal to derivative-- r is just a
constant-- so derivative of the inside is r, times r on
the outside, that's r squared, times e to the rx.
And now we're ready to substitute back in.
And I will switch colors.
So the second derivative, that's r squared times e to
the rx, plus 5 times the first derivative, so that's 5re to
the rx, plus 6 times our function-- 6 times e to the rx
is equal to 0.
And something might already be surfacing to you as something
we can do this equation to solve for r.
All of these terms on the left all have an e to the rx, so
let's factor that out.
So this is equal to e to the rx times r squared, plus 5r,
plus 6 is equal to 0.
And our goal, remember, was to solve for the r, or the r's,
that will make this true.
And in order for this side of the equation to be
0, what do we know?
Can e to the rx ever equal 0?
Can you ever get something to some exponent and get 0?
Well, no.
So this cannot equal 0.
So in order for this left-hand side of the equation to be 0,
this term, this expression right here, has to be 0.
And I'll do that in a different color.
So we know, if we want to solve for r, that this, r
squared plus 5r, plus 6, that has to be 0.
And this is called the characteristic equation.
This, the r squared plus 5r, plus 6, is called the
characteristic equation.
And it should be obvious to you that now this
is no longer calculus.
This is just factoring a quadratic.
And this one actually is fairly
straightforward to factor.
So what is this?
This is r plus 2, times r plus 3 is equal to 0.
And so the solutions of the characteristic equation-- or
actually, the solutions to this original equation-- are r
is equal to negative 2 and r is equal to minus 3.
So you say, hey, we found two solutions, because we found
two you suitable r's that make this
differential equation true.
And what are those?
Well, the first one is y is equal to e to
the minus 2x, right?
We can call that y1.
And then the second solution we found, y2 is e to the--
what is this?-- r is minus 3x.
Now my question to you is, is this the
most general solution?
Well, in the last video, in kind of our introductory
video, we learned that a constant times a solution is
still a solution.
So, if y1 is a solution, we also know that we can multiply
y1 times any constant.
So let's do that.
Let's multiply it by c1.
That's a c1 there.
This is also going to be a solution.
And now it's a little bit more general, right?
It's a whole class of functions.
The c doesn't have to just be 1, it can be any constant.
And then when you use your initial values, you actually
can figure out what that constant is.
And same for y2.
y2 doesn't have to be 1 times e to the minus 3x, it has to
be any constant.
And we learned that in the last video, that if
something's a solution, some constant times
that is also a solution.
And we also learned that if we have two different solutions,
that if you add them together, you also get a solution.
So the most general solution to this differential equation
is y-- we could say y of x, just to hit it home that this
is definitely a function of x-- y of x is equal to c1e to
the minus 2x, plus c2e to the minus 3x.
And this is the general solution of this
differential equation.
And I won't prove it because the proof is fairly involved.
I mean, we just tried out e to the rx.
Maybe there's some other wacko function that would have
worked here.
But I'll tell you now, and you kind of have to take it as a
leap of faith, that this is the only general solution.
There isn't some crazy outside function there that would have
also worked.
And so the other question that might be popping in your brain
is, Sal, when we did first order differential equations,
we only had one constant.
And that was OK, because we had one set of initial
conditions and we solved for our constants.
But here, I have two constants.
So if I wanted a particular solution, how can I solve for
two variables if I'm only given one initial condition?
And if that's what you actually thought, your
intuition would be correct.
You actually need two initial conditions to solve this
differential equation.
You would need to know, at a given value of x,
what y is equal to.
And, maybe at a given value of x, what the
first derivative is.
And that is what we will do in the next video.
See