00:01
Okay, so start off this problem by solving for the homogenous solution.
00:04
So we'll have that r squared minus 5r plus 6 equals to 0.
00:09
And if we factor this, we should have that r minus 2 times r minus 3 equals to 0.
00:18
And this gives us r values of 2 and 3.
00:22
And so with this, we can actually build our homogenous solution.
00:25
So our homogenous solution, y of h, is going to be c1.
00:33
E to the 2x plus c2, e to the 3x.
00:39
And let's recall that the formula, to find a particular solution using variation of parameters, is y2 integral of y1 divided by the ronskin times the right -hand side dx, minus y1 integral of y2 divided by the ronskin times the right -hand side dx.
01:01
And so let's first, define what y1 and y2 is so in this case i'm going to say that this is going to be y1 and this is going to be y2 and so let's start off by finding the ronskin and the ronskin equals to the terminate it's going to use a bracket so the determinant e to the 2x which is y1 and e to the 3x which is y2 and 2 e to the 2x which is y1 prime and 3 e to the 3x which is y1 prime and 3 e to the 3 which is y2 problem.
01:40
And we want to take the determinant of this.
01:42
We're going to have to multiply these two.
01:45
So we're going to get 3e to the 2x times e to the 3x.
01:51
And we're going to have to multiply and subtract these two.
01:54
So subtract 2e to the 2x, e to the 3x.
02:01
And if we multiply two exponentials, we are two powers.
02:05
Two things at the same base.
02:07
We add the exponents.
02:09
So we'll have 3.
02:11
To the 5x minus 2 e to the 5x and so this sum fives down to e to the 5x and so that's our ronskin here and we can actually now build our variation of parameters equation so let's recall that's y2 in this case we said y2 is e to the 3x integral of y1 which is e to the 2x divided by the ronskin e to the 5x times the right -hand side.
02:44
In this case, the right -hand side is 2e to the x, d -x minus y -1, which is e to the 2x, integral of y2, which is e to 3x divided by the ronskin, e to the 5x, times whatever is on the right -hand side, d -x.
03:06
And so let's simplify this down before we try and take on these integrals.
03:14
So we'll have that e to 3x.
03:18
And if we're dividing, we subtract the exponents.
03:21
So this will simplify down to e to negative 3x.
03:27
And i can drag out this 2 here, so i'm going to put it right there.
03:31
And it's going to multiply it by e to the x, dx, minus...
