In Problems 1-5, find a formal solution to the given boundary value problem.
The formal solution to the given initial value problem
Answer to Problem 2E
Solution:
The formal solution to the given initial value problem is
Explanation of Solution
It is given that
With boundary conditions,
and initial conditions,
Consider the solution of the equation be
Above equation satisfies equation (1), so it becomes,
and
Boundary condition in equation (2) becomes,
So, equation (5) becomes,
The auxiliary equation becomes
If
Thus
So, the general solution becomes
By boundary condition,
and
As,
Now, if
The general solution becomes,
By boundary condition,
and
Here
which is given by
Now, put the value of
The auxiliary equation will be,
The solution becomes,
Apply
Now, the boundary conditions in (4) will be satisfied if
Put in equation (7),
Put the value
Then the formal solution obtained is,
where,
The formal solution to the given initial value problem is
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