### All Differential Equations Resources

## 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:**

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 #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:**

There are no solutions to the boundary value problem.

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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