Differential Equations : Second-Order Boundary-Value Problems

Study concepts, example questions & explanations for Differential Equations

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Example Questions

Example Question #1 : Second Order Boundary Value Problems

Find the solutions to the second order boundary-value problem. .

Possible Answers:

There are no solutions to the boundary value problem.

Correct answer:

Explanation:

The characteristic equation of  is , with solutions of . Thus, the general solution to the homogeneous problem is . Plugging in our conditions, we find that , so that . Plugging in our second condition, we find that  and that .

Thus, the final solution is .

Example Question #2 : Second Order Boundary Value Problems

Find the solutions to the second order boundary-value problem. .

Possible Answers:

There are no solutions to the boundary value problem.

Correct answer:

There are no solutions to the boundary value problem.

Explanation:

The characteristic equation of  is  with solutions of . This tells us that the solution to the homogeneous equation is . Plugging in our conditions, we find that  so that . Plugging in our second condition, we have  which is obviously false.

This problem demonstrates the important distinction between initial value problems and boundary value problems: Boundary value problems don't always have solutions. This is one such case, as we can't find  that satisfy our conditions. 

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