00:03
In this video, we're going to go through the solution to question 20 from chapter 4 .5.
00:09
So here we're asked to find the general solution to this in homogenous differential equation.
00:25
So we first find a particular solution to this equation.
00:28
So we take a look at the form of the inhomogeneous part.
00:34
It's a combination of size and causes.
00:39
So we'll let the particular solution.
00:46
Solution as a function of theta be equal to some constant times sine theta plus a different constant times cos theta.
01:10
So we can choose this form because r equals i, as we'll see, is not a root of the auxiliary equation.
01:24
We'll be able to check that later when we go through the home of the general solution to the homogeneous version of this equation.
01:33
So with this we can find the second derivative of the particular solution and this will just be minus alpha, sine theta, minus beta, because the second derivative of sine goes to minus sine, it's active to cause, cause, cause to minus cos.
02:04
So if we substitute that, that into the differential equation, we find that minus alpha plus four times by sine theta plus minus beta plus four times by cos theta is equal to the right inside, which is sign of theta minus cos theta.
02:51
So these bits came from the, yeah, one times here and four times here.
03:05
So then we need to equate the coefficients, because signs and cosies are orthogonal...