00:01
Hi, given the differential equation, triple order, y triple derivative plus 24 y double derivative plus 192 y derivative plus 512y is equal to e to the power minus 8x and we have to solve this using the variation of parameters technique.
00:19
Now auxiliary equation in this case will be mq plus 24m square plus 192m plus 512 equal to 0.
00:30
But x plus 8 whole cube equal to 0 so over here we have x is equal to minus 8 minus 8 now complementary solution in this case will be y is equal to c1 plus c2 x plus c3 x square into e to the power minus 8 x now particular solution will be say y 1 is equal to e to the power minus 8 x y 2 is equal to x e to the power minus 8 x and y 2 is equal to x e to the power minus 8 x and y 3 is equal to x square e to the power minus 8x and my g x is this function g x is this function which is equal to e to the power minus 8x now we find the ronskian so ronskian will be equal to w y 1 y 1 y 2 y 2 y 3 which is equal to y 1 y 1 derivative y 1 double derivative y 2 y 2 derivative y 2 double derivative y3, y3 derivative, y3 double derivative.
01:34
Now we can put the values of y1, y2, y3 and their derivatives respectively and we can solve this.
01:40
So this determinant comes out to be equal to 20 e to the power minus 24x into x square plus 2e to the power minus 24x.
01:54
Now we find w1 so w1 will be equal to now y 1 will become 0 0 and g x and rest all the terms as it is y 3 y 3 derivative y 3 double derivative now you can put all the values and find this so g x is e to the power minus 8x and we can solve this and this comes out to be equal to e to the power minus 24 x into x square similarly we will find w2 and w2 is equal to y 1 y1 derivative y 1 double derivative 0 0 g x this is y 3 y 3 derivative and y 3 double derivative so this will come out to be equal to minus 2 e to the power minus 24 x into x and similarly we can find w 3 so w 3 will be equal to y1, y1 derivative, y1 double derivative, y2, y2 derivative, y2, double derivative, and 0 ,0, jx...