00:01
In this problem, the first question is to solve the equation y double dash minus y is equal to 0 subject to the initial conditions.
00:09
Y of 0 is equal to 0.
00:11
Y dash of 0 is equal to 0.
00:14
Now first consider the given differential equation.
00:16
The corresponding characteristic equation is m square minus 1 is equal to 0.
00:22
Sol the characteristic equation to get the characteristic roots.
00:25
From here we get m square is equal to 1.
00:28
That is m is equal to positive or negative one are the characteristic root see that both are distinct real roots therefore the general solution of the differential equation is y of x is equal to c1 erased to x plus c2 erase to negative x and now we need to use these initial conditions to find the particular solution we have y of zero is equal to zero implies c1 erased to zero plus c2 erase to negative zero is equal to zero that is we get c1 plus c2 is equal to 0 since erased to 0 and 1 by erase to 0 are both 1.
01:05
Now find y dash of x which is the first derivative of this function y of x.
01:10
It is c1 erased to x minus c2 erase to negative x and now y dash of 0 is equal to 0 implies c1 minus c2 is equal to 0.
01:22
Now solving these two equations simultaneously we get 2 c1 is equal to 0 by adding 3.
01:28
These two equations, that is we get c1 is equal to 0 and using the value of c1 we get the value of c2 is also equal to 0...