00:01
Okay, we have three differential equations to solve.
00:05
Here's the first one.
00:08
So the characteristic equation, which is all about the homogenous solution, is r squared plus 5r plus 6, which we're going to set equal to 0.
00:23
It's r plus 3 times r plus 2.
00:27
So r is minus 2 or minus 3, which tells me the characteristic solution looks like that.
00:35
To find a particular solution for this particular equation, it's going to be a polynomial of order x squared, order two.
00:53
Calculate its first and second derivatives, substituted into the equation.
01:05
Okay.
01:06
And then what we're going to want to do is we want to collect terms of the same power of x so the constant term the x term and the x squared term and then we can solve those one at a time this is just a little scratch work i wrote down while i was thinking about how to solve this one so here's my y and there's that answer then the next one is sign for x now again characteristic equation that's y squared minus 2 that should be r not y.
02:33
So we use g instead.
02:39
So g is equal to zero or two.
02:42
Okay.
02:43
So when it's zero, that's not an exponential anymore.
02:47
In fact, the solution is constant for y.
02:56
So there's our characteristic solution.
02:59
The particular solution is a linear combination of a sine and cosine of 4x.
03:12
We have to figure out what alpha and beta are by plugging.
03:15
It into the equation.
03:54
There's our equation.
04:03
Substitute in those derivatives, then collect the terms in sine and cosine...