Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and the equation is called an ordinary differential equation (ODE). If the function depends on several variables, then partial derivatives appear and the equation is called a partial differential equation (PDE). A function that satisfies a differential equation is called a solution of the differential equation. Example: Verify that y 1 (t) = e 3t and y 2 (t) = e t are solutions of the ODE y + 2y 3y = 0. Example: Verify that u(x, t) = e 4t sin x is a solution of the PDE 4u xx u t = 0. 1
Example: Determine the values of r for which the differential equation y 3y + 2y = 0 has a solution of the form y = e rt. Example: Determine the values of r for which the differential equation t 2 y 4ty + 4y = 0 has a solution of the form y = t r for t > 0. 2
Classification of Differential Equations Definition: The order of a differential equation is the order of the highest derivative that appears in the equation. If a differential equation can be written as a linear combination of the unknown function and its derivatives, it is called linear. The general form of a linear ordinary differential equation is a 0 (t)y (n) + a 1 (t)y (n 1) + + a n (t)y = g(t). A differential equation that is not linear is called nonlinear. Example: Classify each differential equation as an ordinary or partial differential equation. Determine the order of each differential equation and whether the equation is linear or nonlinear. (a) (1 + y 2 ) d2 y dt 2 + tdy dt + y = et (b) d3 y dt 3 + tdy dt + (cos2 t)y = t 3 (c) u xx + u yy + u x + 2u = 0 (d) u t + uu x = 1 + u xx 3
Mathematical Models Definition: A differential equation that describes a physical process is called a mathematical model of the process. Example: A tank initially contains 100 L of pure water. A mixture containing 0.2 kg/l of salt enters the tank at a rate of 2 L/min, and the well-mixed solution leaves the tank at the same rate. (a) Find a differential equation for the amount of salt in the tank after t minutes. (b) Find the general solution of this differential equation. The differential equation has infinitely many solutions. The geometric representation of these solutions is an infinite family of curves called integral curves. 4
If we know the initial amount of salt in the tank, then the solution is unique. (c) Suppose that the tank initially contains 100 L of pure water. Find the solution of the differential equation. The condition we used to find the value of C is called an initial condition. The differential equation, together with this initial condition, is an initial value problem. (d) How much salt is in the tank after a long time? 5
Direction Fields and Equilibrium Solutions Consider a differential equation of the form dy dt = f(t, y). The derivative, y, represents the slope of the tangent line to the graph of y(t). By choosing a lattice of points (t, y) in the ty-plane and sketching line segments with slopes y, we obtain a direction field describing the overall behavior of solutions of the equation. Example: Draw a direction field for the differential equation y = y 2. Based on the direction field, determine the behavior of y as t. Definition: An equilibrium solution of a differential equation is a solution from which there is no change. That is, a solution y(t) for which dy dt = y = 0. Solutions may approach or move away from an equilibrium solution over time. 6
Matlab can be used to plot the direction field for a given differential equation along with some of the solution curves. Suppose that we would like to plot the direction field for the differential equation dy dt = 2y + t2 e 2t in the rectangle defined by 0 < t < 5, 3 < y < 3. The following Matlab script can be used to generate the direction field. % Math 308 - Differential Equations % Creates a grid of uniformly spaced points in the rectangle. [T,Y] = meshgrid(0:0.5:5,-3:0.5:3); % Defines the right-hand side of the differential equation. S = 2*Y+T.^2.*exp(-2*T); % Scales the direction field. L = sqrt(1+s.^2); % Plots the scaled direction field. q = quiver(t,y,1./l,s./l,0.5); % Plot information axis tight xlabel( t ) ylabel( y ) 3 2 1 y 0-1 -2-3 0 1 2 3 4 5 t Note: An more detailed version of this script can be downloaded from the course website. 7
Example: Draw a direction field for the differential equation y = 2y 3. Based on the direction field, determine the behavior of y as t. If this behavior depends on the initial value of y at t = 0, describe this dependency. 4 3 2 y 1 0-1 -2 0 1 2 3 4 5 t Example: Draw a direction field for the differential equation y = y(y 5). Based on the direction field, determine the behavior of y as t. If this behavior depends on the initial value of y at t = 0, describe this dependency. y 8 7 6 5 4 3 2 1 0-1 -2 0 1 2 3 4 5 t 8