Section 11.1 What is a differential equation?

Often we have situations in which the rate of change is related to the variable of a function An equation which gives information about the rate of change of an unknown function is called a differential equation Differential equations have functions as solutions –As opposed to numbers (solutions of algebraic equations) Differential equations model more complex problems where the solution is described by a function

Example A yam is placed inside a 200ºC oven. The yam gets hotter at a rate proportional to the difference between its temperature and the oven’s temperature. When the yam is at 120ºC, it is getting hotter at a rate of 2º per minute. Write a differential equation that models the temperature, T, of the yam as a function of time, t.

C is an arbitrary constant In order to solve for C we must be given some kind of initial condition The C in our case is the initial temperature difference between the yam and the oven What is C if the initial temperature of the yam is 20º?

Family of solutions for different values of C

Antidifferentiation is actually a particular case of solving a differential equation, in particular Where the constant appears added to the solution, not multiplied The solution is a function and unless specific conditions are given there are usually many solutions to a certain differential equation The way to tell if a given function is a solution to a certain differential equation is by substitution

Examples Verify that the following are solutions to the given differential equation

Second-order differential equations A second-order differential involves the second derivative This involves two antidifferentiations so it will involve two arbitrary constants For example let’s look at the second-order differential equation