Study Guide

Differential Equations - Differential Equations and Their Solutions

Differential Equations and Their Solutions

A differential equation (d.e.) is any equation that has one or more derivative in it. These can be first derivatives, second derivatives...whatever.

Sample Problem

The following are differential equations.

' + " + xy = 0

Sample Problem

The following are not differential equations, because they don't contain any derivatives.

x2 + y2 = 8

x + xy â€“ y + 9 = 0

x = 9

The order of a differential equation is the highest derivative that occurs in that differential equation.

Sample Problem

The differential equation

' + " + "' + x = 0

has order 3 because that's the highest derivative in the equation:

' + " + "' + x = 0.

Sample Problem

The differential equation

has order 1 because it only contains a first derivative.

A d.e. of order 1 is called a first-order differential equation, and a d.e. of order 2 is called a second-order differential equation. These are the kinds of differential equations that you'll probably see most often.

  • Solutions to Differential Equations

    Way back in algebra we learned that a solution to an equation is a value of the variable that makes the equation true. This is backwards kind of thinking we need for differential equations.

    To check if a number is a solution to an equation, we evaluate the left-hand side of the equation at that number, then evaluate the right-hand side of the equation at that number. If we get the same value for both sides of the equation, the number is a solution to the equation.

  • Solving Differential Equations

    Checking to see if a given function satisfies a given differential equation isn't too horrible of a task (at least for the functions we encounter in Calculus).

    The task of solving differential equations from scratch is a bit different.

    Differential equations fall into several very, very broad categories:

    • ones you can solve right now by thinking backwards.
        
    • ones you'll be able to solve by the end of the calculus class.
        
    • ones you'll be able to solve if you take other courses about differential equations, and
        
    • ones that people at places like MIT are still working on.

    We need to ignore most of these for now and concentrate on the ones we can solve by thinking backwards.

  • Initial Value Problems

    An Initial Value Problem (IVP) is a differential equation combined with one or more initial conditions. An initial condition gives some extra information about the solution. In order to be a solution to an IVP, a function has to satisfy both the differential equation and all initial conditions.

    This will allow to find the value of that pesky constant we're always adding onto our solutions, so that we can get a unique solution.

  • More About Solutions

    Differential equations have two kinds of solutions: general and particular.

    The general solution to a differential equation is the collection of all solutions to that differential equation. A general solution will usually contain some undetermined constants.

    Sample Problem

    y = 4x + C is the general solution to the d.e.

    ' = 4.

    Sample Problem

    y = 3x2 + Bx + C is the general solution to the d.e.

    " = 6.

    A particular solution to a differential equation is a solution with all the constants filled in.

    Sample Problem

    The function y = 4x + 2 is a specific solution to the d.e.

    ' = 4.

    Initial value problems usually have a particular solution only, because the initial condition(s) force us to pick values for the constant(s) in the general solution.