00:01
The first question is y w -dash minus 2 y -dash plus y -d is equal to x square minus 3x plus sign of 2x.
00:16
Now let us use the method of undetermined coefficients.
00:19
And also it's given that y of 0, y -dash of 0 is equal to 1.
00:25
So let's solve this problem first.
00:27
So first of all, the homogeneous counterpart of this is d square minus 2d plus 1 is equal to 0.
00:34
The roots are obviously both are equal.
00:36
So the complementary solution, yc of x, is a x plus b into e par x.
00:45
But a and they are arbitrary constants.
00:50
Now by method of undetermined coefficients, my y p of x will be of the form lambda x squared plus mu x plus some t because this is a quadratic and taking quadratic and there is one trigonometric part also plus some alpha sine 2x plus beta cost 2x so this is my y p of x now let's differentiate yp of x is equal to 2 lambda x plus mu plus 2 alpha cost 2 x minus 2 beta sine 2x once more yp double dash will be 2 lambda minus 4 alpha sine 2 x minus 4 beta cost 2x so now let us plug in yp double dash yp dash and yp in this because every yp will satisfy the original differential equation so we have yp double dash that is 2 lambda minus 4 alpha sine 2 x minus 4 beta cost 2x this is y to p double as minus two times of yp dash so two times let's multiply here so it is 4 lambda x minus 2 mu minus 4 alpha cost 2 x minus 2 into minus 2 beta it's plus 4 beta sine 2 x so this is minus 2 yp dash and plus y p that is plus lambda x square plus mu x plus t plus alpha sine 2x plus beta cost 2x this should be equated to x square minus t x plus sine 2x plus 1 sine 2x with 0 cost 2x plus some constant 0 so what is the reason we are doing this is we can compare the coefficients so what is the coefficient of x is square you see only lambda is the coefficient of x squared so lambda is one.
03:05
Next what is the coefficient of x? so that you have to do it carefully.
03:09
It is minus 4 lambda, minus 4 lambda and plus mu and that should be is equal to negative 3.
03:16
But just now we got it lambda, lambda is 1.
03:18
So mu is equal to 1.
03:22
Next the constant, the constant is 2 lambda and minus 2 mu and t...