00:01
Hello everyone, in this question we have given a differential equation that is y double dash minus 9y this is equals to 9x divided by e to the power 3x and we need to solve this by using variation of parameter.
00:12
So for that now for the complementary function let's make the auxiliary equation that is m square minus 9 this is equals to 0 and we have roots that is clearly 3 and minus 3.
00:25
Right.
00:25
So clearly we can write complementary function as that is c1 e to the power 3x and plus c2 e to the power minus of 3x.
00:35
Right.
00:35
Now clearly our independent solutions are e to the power 3x and e to the power minus 3x.
00:43
Now for the particular integral we know that y of p is equals to u1 y1 plus u2 y2 and clearly u1 is equals to we have minus of y2 divided by wronskian times of f of x and we need to integrate this with respect to x and y u2 is basically integral of this is integral of y1 divided by w and we have f of x dx.
01:18
Right.
01:18
So clearly in order to find that we need to find wronskian first.
01:23
Right.
01:24
So for wronskian we know the formula that is determinant of y1 y2 and then y1 dash y2 dash.
01:31
Right.
01:32
So y1 is basically e to the power 3x and y2 is e to the power minus 3x when we differentiate it we get 3 times of e to the power 3x minus 3 times of e to the power minus 3x.
01:44
Right.
01:45
So when we solve this we get minus 6 which is not is equals to 0.
01:48
Right.
01:49
So further we have u1 is equals to minus as it is now y2 that is e to the power minus 3x divided by wronskian that is minus 6 and what is f of x.
01:59
So basically f of x is this 9x divided by e to the power 3x and we need to integrate with respect to x.
02:08
So we have 1 divided by 6 as it is we can take outside 9 and we have x e to the power minus 6x dx.
02:18
So this is equals to 3 divided by 2 and we have x e to the power minus 6x dx.
02:25
Right.
02:25
Now we can integrate this by using integration of parts integration by parts.
02:30
Right.
02:31
So clearly first function that is x as it is and integral of second that is e to the power minus 6x divided by minus 6 minus the differentiation of first that is 1 and again the integral of e to the power minus 6x and then whole integral...