00:01
For this problem we're asked to solve this differential equation, and i'm going to do that by using the method of undetermined coefficients.
00:06
So first i'm going to solve the homogeneous differential equation.
00:10
That is, the right -hand side is set equal to zero.
00:14
So the auxiliary equation is r minus one, this is equal to zero.
00:20
And then obviously r, this is equal to one.
00:22
So the complementary solution is some constant, call it c1, times e to the t.
00:29
And just as a reminder of the general solution, this is equal to the complementary plus the particular.
00:36
So since we have the complementary, we can make our guess for the particular.
00:41
And it's influenced by this function right here, and then also by the complementary solution.
00:47
So just looking at this function, we would guess a times e to the t, plus b times t, plus c.
00:58
Where a, b, and c are all constants.
01:01
However, this function is also seen in the complementary, and we can't have any overlap, so we'll multiply this by t.
01:10
Now there's no overlap between the complementary and the particular.
01:14
So now we'll take this function, we'll plug it into this differential equation, forcing it to be a solution, and then we'll solve for the constants a, b, and c.
01:23
And in order to do that, first i need the derivative of y sub p.
01:26
So this is a e to the t plus a t, e to the t.
01:34
I can use the product rule to calculate that.
01:37
And then now i'm plugging in the particular solution to the differential equation.
01:42
So i have t minus 2 e to the t...