00:01
In this question it is given that 2 d square y by d t square negative 4 d y by d t plus y is equal to 4 t square also it is given that y 0 is equal to 5 and y 10 is equal to 10.
00:32
Consider the given the given equation as 2 y double dash negative 4 y dash plus y is equal to 4 t squared now we first find the complementary solution so for this we need to find the auxiliary equation that will be equals to 2 m square negative 4m plus 1 so here we'll find the value of m as m is equals to negative b positive negative square root b square negative 4 a c divided by 2a so here in this equation a is equal to 2 b is equal to negative 4 and c is equal to 1 so these are the coefficients of this auxiliary equation now we'll put the value of a b c in this and we get m is equals to 4, positive negative, 16 negative 8 divided by 4.
01:54
Now on solving it further, we get 4 positive negative square root 8 divided by 4.
02:05
Now on solving it, it can be written as 1, positive negative 1 divided by root 2.
02:21
Now we get the two roots as m is equals to 1 plus 1 divided by 2 and the other root is 1 negative 1 divided by root 2.
02:39
So here we have the solution as y is equals to c1 e raise to the power 1 positive 1 divided by root 2 t plus c2 e raise to the power.
02:50
1 positive 1 divided by root 2 t plus c2 e raise to the power.
02:56
Are 1 negative 1 by root 2 t so now we need to find the particular solution so for this we have 1 divided by 2 d square negative 4d plus 1 and here we have 4 t square now here this can be written as 1 divided by 1 plus 2 d square negative 2 d square negative 4d here we have 4 t square now note that here this forms the expansion of 1 divided by 1 negative z that is equals to summation of z raised to the power n where n will be from 0 to infinity so here its expansion will be in the form of 1 plus z plus z square up to so on so now here here this thing can be written as 1 divided by 1 negative...