00:01
In this question, we will be talking about linear differential equations.
00:06
In particular, we will be solving the following first order linear differential, er, sorry, initial value problem, which is just a differential equation with an initial condition.
00:20
So first we solve this de, and then we will apply the initial condition.
00:28
How do we solve first order linear differential equations? well we use the method we can use the method of integrating factors whose goal is to get the left side to be of the form derivative with respect to the independent variable of a function of the independent variable times y the dependent variable now the variables can have different names like r or t or s or whatever whatever letters you choose, but remember that it doesn't matter that it's y and x, it's just the dependent variable and the independent variable.
01:17
Okay, so anyways, the goal of the method of integrating factors is to write the left side as this, in this form, and the right side as just another function of the independent variable, in this case, x.
01:33
So this method gives us a general way of finding a function, of x by which we multiply both sides of our equation to get the left side to be of this form, derivative with respect to x of f of x times y.
01:54
However, always check your left side, well, if you have all the dependent variable terms and dependent variable derivative terms on the left side, check if it is already of that form, because then you don't have to find f of x.
02:09
It's already multiplied on the standard form of the differential equation.
02:17
So let's check this equation.
02:21
So we have a function of x times the derivative of y plus the derivative of that function of x times y.
02:33
And that's the exact statement of the derivative of this function of x times y.
02:42
Function of x here is just that x.
02:49
The right side remains the same, and so this is our equation now.
02:57
Once we have it in this form, it's easy to solve, since all we have to do to get rid of the integration is integrate both sides with respect to x.
03:12
The left side is the antiderivative of the derivative of x times y, which is just x times y, and the right side is an integral that it's not so easy to solve, but it can be done using integration by parts.
03:32
So remember the formula of four integration by parts.
03:42
Consider the two functions of x, f and g in your integral, where the derivative of f and g multiplied are being integrated.
03:55
That is equal to f times g minus the integral of f times g prime, the derivative of g.
04:04
So let's consider this x a derivative of some function, because we can undo that differentiation quite easily, since x is a polynomial.
04:14
And let's consider long x, the function g, in our case.
04:22
And now applying in the formula, we write f times g.
04:27
Remember, g is just lax, and f will be the antiderivative of f prime, which is x, and we know that to be one half x squared...