00:01
In this question, the fourier cosine series, since f of t is an odd function, f of minus t is equal to minus f of t.
00:10
The fourier cosine series coefficient will be 0.
00:16
And now the fourier sine series, the general formula for the fourier sine series coefficient is bn is equal to 2 by l integral of f of t into sine n pi t by l into dt.
00:39
So, l is the period of the function.
00:42
So, in this case, l is equal to 2 and we have bn is equal to 2 by 2 integral 1 minus t into sine n pi t by 2 into dt, which is equal to integral of 1 minus t sine n pi t by 2 into dt is equal to minus 2 by n pi t whole square into cos n pi minus cos 0, which is equal to 4 by n pi square 1 minus minus 1 d whole power n.
01:27
Therefore, the fourier sine series of f of t is f of t is equal to summation bn sine n pi t by 2 from n is equal to 1 to infinity, which is equal to we get 4 by pi sine pi t by 2 minus 1 by 3 sine 3 pi t by 2 plus 1 by 5 into sine 5 pi t by 2 minus.
02:10
And the end point value problem, the characteristic equation of the differential equation is x double dash plus 2x is equal to 0 is r square plus 2 is equal to 0, which has the roots r is equal to plus or minus root 2 of 5.
02:28
Therefore, the general solution of the homogeneous equation is xh of t is equal to c1 cos root 2t plus c2 sine root 2t.
02:40
To find a particular solution of the non -homogeneous equation, we can use the method of undetermined coefficients...