Section 1.1. Introduction to Differential Equations

Objectives:

1. The definition of differential equations

2. The real applications to the differential equations

Consider the equation [latex]y'=3x^{2}[/latex], which is an example of a differential equation because it includes a derivative. There is a relationship between the variables [latex]x[/latex] and [latex]y[/latex]: [latex]y[/latex] is an unknown function of [latex]x[/latex]. Furthermore, the left-hand side of the equation is the derivative of [latex]y[/latex]. Therefore we can interpret this equation as follows: Start with some function [latex]y=f(x)[/latex] and take its derivative. The answer must be equal to [latex]3x^{2}[/latex]. What function has a derivative that is equal to [latex]3x^{2}[/latex]? One such function is [latex]y=x^{3}[/latex], so this function is considered a solution to a differential equation.

Definition: A differential equation is an equation involving an unknown function [latex]y=f(x)[/latex] and one or more of its derivatives. A solution to a differential equation is a function [latex]y=f(x)[/latex] that satisfies the differential equation when [latex]f[/latex] and its derivatives are substituted into the equation.

Real world examples of differential equations

Example 1: The rate of change of population with respect to time is proportional to its current population with initial population given: [latex]y'=2y[/latex], [latex]y(0)=100[/latex] where [latex]y[/latex] is the population function with respect to time [latex]t[/latex]. 

Exercise 1: If the rate of change of the owl population with respect to time is 1.5 times to it current population with initial population of 1000 owls. Write a differential equation model to present the owl population. 

Example 2: The rate of change of the concentrate of the solution in a container is the rate of the change of of the concentrate of the solution flow into the container subtracted by the rate of change of the concentrate of the solution flow out of container.

Exercise 2: Let [latex]Q[/latex] g be the amount of salt in the salt lake. If 2 g/L of salt water is flowing into the salt lake with the rate of 3L per hour and 1 g/L of salt water is flowing out of the lake of 3L per hour. Write a differential equation for the rate of change of the salt in the lake with respect to the time [latex]t[/latex] hour. 

Example 3: The net force on a spring is the sum of negative proportional to its current position, negative proportional to its current velocity, and external force: [latex]my''=-ry'-ky+F(t)[/latex], where [latex]y[/latex] is the position function with respect time [latex]t[/latex].

Example 4: The heat conduction problem that involves two valuables: [latex]au_{xx}=u_{t}[/latex] where [latex]u(x,t)[/latex] is the temperature function of a rod depending on the position and the time.

Group work

1. Verify [latex]y=100e^{2t}[/latex] is a solution of [latex]y'=2y[/latex], [latex]y(0)=100[/latex].

2. Verify [latex]y_{1}=e^{2t}[/latex] and [latex]y_{2}=e^{3t}[/latex] are solutions of [latex]y''-5y'+6y=0[/latex]. Show for any [latex]c_{1}[/latex] and [latex]c_{2}[/latex] constant that [latex]y=c_{1}e^{2t}+c_{2}e^{3t}[/latex] is a solution of [latex]y''-5y'+6y=0[/latex]. 

3. Verify [latex]y_{1}(t)=t/3[/latex] and [latex]y_{2}(t)=e^{-t}+t/3[/latex] are solutions of [latex]y''''+4y'''+3y=t[/latex].