00:01
Hello students, here we have to solve this given differential equation using the series method.
00:06
So, let us consider the series y equal to summation n equal to 0 to infinity a n x bar n and by differentiating this we will get y dash equal to summation n equal to 0 to infinity n a n x bar n minus 1 and y double dash will be summation n equal to 0 to infinity n into n minus 1 into a n x power n minus 2.
00:42
Here we can rewrite this as summation n equal to 2 to infinity by substituting 0 for n we will get the term 0 and 1 for n we will get the term 0 such that we can consider n equal to 2 to infinity n into n minus 1 into a n x power n minus 2.
01:05
Here the first few terms of the solution are given that y equal to a naught plus a 1 x plus a 2 x square plus a 3 x 2 plus a 4 x power 4.
01:15
Here we have to find the coefficients.
01:17
First we have to substitute the values of y y dash and y double dash in the given differential equation that is by substituting the values we will get the equation like this.
01:34
Then we have to take the coefficients term and x power n terms therefore we will get 2 a 2 plus a naught plus summation n equal to 1 to infinity x power n as common we will get n plus 2 into n plus 1 into a n plus 2 plus 3 n a n plus a n equal to 0.
01:53
Here we have to equate the coefficients of like powers therefore we will get 2 a 2 plus a naught equal to 0 and this term also equal to 0 therefore we will get a 2 equal to minus 1 by 2 a naught and a n plus 2 equal to minus 3 n plus 1 by n plus 2 into n plus 1...