Let's solve another 2nd order linear homogeneous
differential equation.
And this one-- well, I won't give you the details before I
actually write it down.
So the differential equation is 4 times the 2nd derivative
of y with respect to x, minus 8 times the 1st derivative,
plus 3 times the function times y, is equal to 0.
And we have our initial conditions y of
0 is equal to 2.
And we have y prime of 0 is equal to 1/2.
Now I could go into the whole thing y is equal to e to the
rx is a solution, substitute it in, then factor out e to
the rx, and have the characteristic equation.
And if you want to see all of that over again, you might
want to watch the previous video, just to see where that
characteristic equation comes from.
But in this video, I'm just going to show you, literally,
how quickly you can do these type of problems mechanically.
So if this is our original differential equation, the
characteristic equation is going to be-- and I'll do this
in a different color-- 4r squared minus 8r plus 3r is
equal to 0.
And watch the previous video if you don't know where this
characteristic equation comes from.
But if you want to do these problems really quick, you
just substitute the 2nd derivatives with an r squared,
the first derivatives with an r, and then the function
with-- oh sorry, no.
This is supposed to be a constant--
And then the coefficient on the original function is just
a constant, right?
I think you see what I did.
2nd derivative r squared.
1st derivative r.
No derivative-- you could say that's r to the 0, or just 1.
But this is our characteristic equation.
And now we can just figure out its roots.
This is not a trivial one for me to factor so, if it's not
trivial, you can just use the quadratic equation.
So we could say the solution of this is r is equal to
negative b-- b is negative 8, so it's positive 8-- 8 plus or
minus the square root of b squared.
So that's 64, minus 4 times a which is 4,
times c which is 3.
All of that over 2a.
2 times 4 is 8.
That equals 8 plus or minus square root of 64 minus--
what's 16 times 3-- minus 48.
All of that over 8.
What's 64 minus 48?
Let's see, it's 16, right?
Right.
10 plus 48 is 58, then another-- so it's 16.
So we have r is equal to 8 plus or minus the square root
of 16, over 8, is equal to 8 plus or minus 4 over 8.
That equals 1 plus or minus 1/2.
So the two solutions of this characteristic equation--
ignore that, let me scratch that out in black so you know
that's not like a 30 or something-- the two solutions
of this characteristic equation are r is equal to--
well 1 plus 1/2 is equal to 3/2-- and r is equal to 1
minus 1/2, is equal to 1/2.
So we know our two r's, and we know that, from previous
experience in the last video, that y is equal to c times e
to the rx is a solution.
So the general solution of this differential equation is
y is equal to c1 times e-- let's use our first r-- e to
the 3/2 x, plus c2 times e to the 1/2 x.
This differential equations problem was literally just a
problem in using the quadratic equation.
And once you figure out the r's you have
your general solution.
And now we just have to use our initial conditions.
So to know the initial conditions, we need to know y
of x, and we need to know y prime of x.
Let's just do that right now.
So what's y prime?
y prime of our general solution is equal to 3/2 times
c1 e to the 3/2 x, plus-- derivative of the inside-- 1/2
times c2 e to the 1/2 x.
And now let's use our actual initial conditions.
I don't want to lose them-- let me rewrite them down here
so I can scroll down.
So we know that y of 0 is equal to 2, and y prime of 0
is equal to 1/2.
Those are our initial conditions.
So let's use that information.
So y of 0-- what happens when you substitute x
is equal to 0 here.?
You get c1 times e to the 0, essentially, so that's just 1,
plus c2-- well that's just e to the 0 again, because x is
0-- is equal to-- so this is, when x is equal to 0, what is
y? y is equal to 2.
Y of 0 is equal to 2.
And then let's use the second equation.
So when we substitute x is equal to 0 in the derivative--
so when x is 0 we get 3/2 c1-- this goes to 1 again-- plus
1/2 c2-- this is 1 again, e to the 1/2 half times 0 is e to
the 0, which is 1-- is equal to-- so when x is 0 for the
derivative, y is equal to 1/2, or the derivative is 1/2 at
that point, or the slope is 1/2 at that point.
And now we have two equations and two unknowns, and we could
solve it a ton of ways.
I think you know how to solve them.
Let's multiply the top equation-- I don't know--
let's multiply it by 3/2, and what do we get?
We get-- I'll do it in a different color-- we get 3/2
c1 plus 3/2 c2 is equal to-- what's 3/2 times 2?
It's equal to 3.
And now, let's subtract-- well, I don't want to confuse
you, so let's just subtract the bottom from the top, so
this cancels out.
What's 1/2 minus 3/2?
1/2 minus 1 and 1/2.
Well, that's just minus 1, right?
So minus c2 is equal to-- what's 1/2 minus 3?
It's minus 2 and 1/2, or minus 5/2.
And so we get c2 is equal to 5/2.
And we can substitute back in this top equation.
c1 plus 5/2 is equal to 2, or c1 is equal to 2, which is the
same thing as 4/2, minus 5/2, which is equal to minus 1/2.
And now we can just substitute c1 and c2 back into our
general solution and we have found the particular solution
of this differential equation, which is y is equal to c1-- c1
is minus 1/2-- minus 1/2 e to the 3/2 x plus c2-- c2 is
5/2-- plus c2, which is 5/2, e to the 1/2 x, and we are done.
And it might seem really fancy.
We're solving a differential equation.
Our solution has e in it.
We're taking derivatives and we're doing
all sorts of things.
But really the meat of this problem was solving a
quadratic, which was our characteristic equation.
And watch the previous video just to see why this
characteristic equation works.
But it's very easy to come up with the characteristic
equation, right?
I think you obviously see that y prime turns into r squared,
y prime turns into r, and then y just turns into 1,
essentially.
So you solve a quadratic.
And then after doing that, you just have to take one
derivative-- because after solving the quadratic, you
immediately have the general solution-- then you take its
derivative, use your initial conditions.
You have a system of linear equations which is Algebra I.
And then you solve them for the two constants, c1 and c2,
and you end up with your particular solution.
And that's all there is to it.
I will see you in the next video.