Exam Questions - Particular solutions using boundary conditions

Exam Questions - Particular solutions using boundary conditions

Particular Solutions using Boundary Conditions

Understanding Boundary Conditions

• Boundary conditions are specific input or output values assigned for a differential equation.
• These conditions are given for the extremities (boundary) of the possible solutions.
• Boundary conditions are used to select the solution that meets the physical requirements of the problem.

The General and Particular Solution

• Generally, the solution to a homogeneous second-order differential equation is given in a form of general solution.
• This general solution may include arbitrary constants.
• To find a particular solution, you can substitute the given boundary conditions into the general solution to obtain particular values for these constants.

Working with Particular Solutions

• Identify the general solution: note that this always includes one or more arbitrary constants.
• Substitute the given boundary conditions into the general solution. There will typically be as many equations as there are arbitrary constants.
• Solve these simultaneous equations to determine the values of the constants.
• Substitute these values back into the general solution, resulting in the particular solution which satisfies the boundary conditions.

Recognising Types of Boundary Conditions

• Initial conditions specify values for the solution and possibly its derivatives at a certain point.
• Boundary value conditions provide values at two different points.
• Periodic boundary conditions suggest that the function repeats after a certain period.

Techniques

• Knowledge of algebraic techniques including factorisation, fractions and roots are essential.
• Basic understanding of integration and differentiation is required.
• Ability to solve simultaneous equations is necessary for finding the values of constants.