General solution and Particular Solution of a Differential Equation Video Lecture | Mathematics (Maths) Class 12 - JEE

In this video we are going to talk about
the
concept of a general solution of a differential
equation and how it compares to the concept of
a particular solution of a differential equation.
At the end, we'll use these ideas to solve an
exercise.
Let's start with the differential equation dy dt
equals 2 y as an example.
It turns out that for this equation the general
solution is y equals C e to the 2 t.
We came up with this solution in an earlier
video, but it's okay if you haven't watched that
yet.
What we want to do here is to understand what
it means to say that this solution is the general
solution.
So far we have a formula for the solution, which
represents an algebraic perspective.
To get a better handle on this, let's look next at
a geometric perspective.
In other words, let's look at the graph of the
solution.
To make the graph we have to realize that the
constant C could represent any real number.
So, we're going to get not just one curve, but
instead, infinitely many curves.
We can start by picking one value of C,
say, C equals one, and then we do just get one
curve.
In this case, y equals one times e to the 2 t, or
just e to the 2 t.
The graph of that function looks approximately
like
this.
Next we can choose a different value of C, say
two, and we get a different curve, which looks
roughly like
this.
We can continue in this way even choosing
fractional or negative values of C, for instance,
and each time we choose a different value, we'll
get a different curve.
Keeping that in mind, I'm just going to plot a
bunch of different curves to give you a better
sense of the larger picture.
Okay, let's complete our understanding with a
verbal description of the general solution.
The essential idea is that a general solution is
not just any family of solutions, but instead
it actually includes all possible solutions
and no functions that are not solutions.
Alright, we've discussed what it means to say
that a solution is the general solution.
Next, let's discuss what it means to say that a
solution is a particular solution.
To understand this, let's look at the same
differential equation as before:
dy dt equals two y, and to get an example of a
particular solution, we just choose one
particular value of C.
So, the example we chose before, where C was
equal to one, was actually an example of a
particular solution.
In that case we had the particular solution y
equals e to the 2 t.
If we were to choose a different value of C, say C
equals negative 1/3, we would get a different
particular solution.
Now, the graph of a particular solution is just a
single curve,
which if C equals one would look approximately
like that.
In other words, a particular solution is just one
individual solution whose graph is a single curve,
which you might remember is called an integral
curve.
Okay, to wrap up this video, let's finish with a
quick exercise were we actually find the general
solution for a certain differential
equation and then use that to determine the
particular solution that satisfies a given
condition.
Through this example, we'll also see a
connection between what we're learning now and
what you learned in calculus.
Here's the exercise. Suppose that y prime is
equal to one over the quantity one plus t
squared,
and we want to (a) Determine the general
solution y,
and (b) Determine the particular solution
satisfying y of one equals zero. The solution to
part (a) is relatively straightforward.
Since we're given the derivative of y as a function
of t, we can just integrate both sides to get that
y is equal to the integral of one over
the quantity of one puts t squared
dt, and you might remember from calculus that
this is an integral that comes up fairly frequently.
Fortunately, we have a nice formula for it, which
is arctangent of t plus C.
And that's it! That is the general solution.
Before we move on to part (b) I just want to
make a quick note, which is that in this
example, the general solution
is just the general antiderivative. So, when you
found antiderivatives in calculus
you were actually solving basic differential
equations.
Okay, let's go ahead and determine the solution
to part (b) which is equally
straightforward. We just need to find the value of
C that will give us a solution
satisfying y of one equals zero, and to do that,
we substitute the information into the formula for
our solution,
and we know that y would have to equal zero
when t is equal to one
and that enables us to solve for C.
You just have to understand that the arctangent
of one is pi/4
and from there we get that C is equal to negative
pi/4, and therefore, y must equal
arctangent of t minus pi/4, and this is the
particular solution satisfying
y of one equals zero.
Alright, that's it for now and I'll see you in the
next video!