Find all the real eigen-values and eigen-functions for the given eigen value problem.
a.
b.
(a)
To find:
All the real eigenvalues and eigenfunctions for the eigenvalue problem
Answer to Problem 1RP
Solution:
The real eigenvalues are
Explanation of Solution
Calculation:
The given eigenvalue problem is,
The boundary values are given as,
The auxiliary equation of equation
The nature of roots of equation
Consider,
Let
Equation
Roots of the above equation will be given as,
The solution is given as follows.
Since,
Again, since
Thus,
This implies, there is no solution.
Now, consider,
Equation
Roots of the above equation will be given as,
The solution is given as follows.
Since,
Again, since
Thus,
This implies, there is no solution.
Now, consider,
Let
Equation
Roots of the above equation will be given as,
The solution is given as follows.
Since,
Again, since
Since,
Thus,
Here,
The real eigenvalues are as follows,
Substitute
The real eigenfunctions are given as follows,
Here,
Therefore, the real eigenvalues are
Conclusion:
Hence, the real eigenvalues are
(b)
To find:
All the real eigenvalues and eigenfunctions for the eigenvalue problem
Answer to Problem 1RP
Solution:
The real eigenvalues are
Explanation of Solution
Calculation:
The given eigenvalue problem is,
The boundary values are given as,
The auxiliary equation of equation
The nature of roots of equation
Consider,
Let
Equation
Roots of the above equation will be given as,
The solution is given as follows.
Since,
Again, since
Since,
Thus,
This implies, there is only one solution, having positive value to this problem and it is denoted by
Here,
Substitute
Thus, corresponding eigenfunctions are,
Now, consider,
Equation
Roots of the above equation will be given as,
The solution is given as follows.
Since,
Again, since
since,
This implies, there is no solution.
Now, consider,
Let
Equation
Roots of the above equation will be given as,
The solution is given as follows.
Since,
Again, since
Since,
Thus,
The eigenvalues are as follows,
Here,
Substitute
The real eigenfunctions are given as follows,
Here,
Therefore, the real eigenvalues are
Conclusion:
Hence, the real eigenvalues are
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Chapter 11 Solutions
Fundamentals of Differential Equations and Boundary Value Problems
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