Particular solutions using boundary conditions to solve differential equations

Understanding Particular Solutions and Boundary Conditions

A differential equation is a mathematical equation involving derivatives of a function. It describes the rate at which something changes.
The general solution of a differential equation describes every possible function or curve that fits the equation.
A particular solution to a differential equation is a specific curve that satisfies the equation and also fulfils a set of additional conditions, known as boundary conditions.
Boundary conditions are supplementary conditions which the solution must satisfy. They are usually given as function values at specific points.

To find the particular solutions to a differential equation, you must first find the general solution.
Next, apply the boundary conditions to solve for any constants in the general solution.
Substituting the values provided by the boundary conditions into the equations should allow you to solve for constants, hence providing the particular solution.
First order differential equations require a single boundary condition, while second order differential equations often need two.

Always start by finding the general solution to the differential equation.
Watch carefully for the clarifying information in boundary conditions. They provide specific value of the function or its derivative at certain points.
For equations with multiple constants, additional boundary conditions are typically necessary. The number and type of these will vary depending on the order of the differential equation.
Practice is key to mastering this technique. Ensure to get to grips with the process by trying many different equations with different orders and boundary conditions. Use diagrams and graphs if necessary as they can often help your understanding of what’s happening mathematically.
Despite the complexity of some boundary conditions, remember that they are fundamentally tools to help you find the particular solution from the general solution. If you’re getting stuck, it can often help to return to first principles: solve the differential equation and apply the boundary conditions.