Differential Equations and General and Particular Solutions

Differential Equations

Did You Know? Differential equations began with Leibniz, the Bernoulli brothers, and others from the 1680s, not long after Newton's fluxional equations in the 1670s. As introduced in the Integration module, an equation containing one or more derivatives of a function is a differential equation. The presentation below defines a differential equation.

The order of a differential equation is the ordinal number of the highest derivative that occurs in a differential equation. For example, a first order differential equation contains a first derivative; a second order differential equation contains a second derivative; etc. View a presentation introducing differential equations.

General and Particular Solutions

A solution to a differential equation is a differentiable function f defined on an interval I of x-values such that when y = f(x) and its derivative(s) are substituted into the differential equation, the resulting equation is true for all x over the interval I. Solving a differential equation involves finding all possible solutions to the equation and is not always a simple task. The general solution of a differential equation is a family of solutions of a differential equation and always contains an arbitrary constant. Recall that solving a first order differential equation such as equation image indicator $LaTeX: \frac{dy}{dx}=f\left(x\right)$ produces a family of functions y = F(x) + C that make the equation true. Solving a simple differential equation is illustrated below.

In most situations involving differential equations, it is not the general solution that is of interest, but rather the particular solution , the one solution of a differential equation that satisfies a condition of the form y(x₀) = y₀ . Recall from the Integration module that if you know the value of y = F(x) for one value of x, namely x₀, then it is possible to find a particular solution of an initial-value problem that satisfies an initial condition. The presentation below illustrates how to find the particular solution to a basic differential equation.

Verification of solutions of differential equations is illustrated in the next presentation.

Differential Equations and General and Particular Solutions Practice

Differential Equations and General and Particular Solutions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of differential equations and general and particular solutions.

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