Section 2.1 Differential Equation and Solutions Key Terms: Ordinary Differential Equation (ODE) Independent Variable Order of a DE Partial Differential Equation (PDE) Normal Form Solution General Solution Solution Curves (Integral Curves) Initial Value Problem (IVP) Particular Solution Interval of Existence Direction Field Euler s Method Quantitative Methods Equilibrium Points Equilibrium Solution
Definition: An ordinary differential equation is an equation involving an unknown function of a single variable together with one or more of its derivatives. dp() t dt = rp() t Population model dp() t P() t = r 1- P() t dt K Logistic population model dx() t x() t = 2 - ( 52 + 20cos(2 π(t - 1))) ) dt 100 Pollution model Each of these ODEs has independent variable t. On occasion we will have the unknown function y(x), where x is the independent variable or even y(θ) where θ is the independent variable. (The independent variable is the symbol we differentiate with respect to.) Definition: The order of a differential equation is the order of the highest derivative that occurs in the equation. Each of the DEs above is a first order ODE.
Definition: Every ODE can be rewritten so it is in normal form.
Example: ODE 6yy ꞌ cos(t) = 0 has normal form y' = cos(t) 6y To put an ODE in normal form just solve the highest order derivative. Definition: A solution of the first-order ordinary differential equation φ(t, y, yꞌ) = 0 is a differentiable function y(t) such that φ(t, y(t), yꞌ(t)) = 0 for all t in the interval where y(t) is defined. To discover if a given function is a solution to a differential equation we substitute the function and its derivative(s) into the equation. You can use this procedure to check that your homework solutions are correct.
Example: -t 2 -t 2 y' = Ce (-2t) and - 2ty = -2tCe This example illustrates the fact that a DE can have lots of solutions. The solution formula -t 2 y = Ce gives a different solution for every value of the constant C. In Section 2.4 we show that every solution to DE yꞌ = -2ty is of this form for some value of the constant C. For this reason the formula is called the general solution to yꞌ = -2ty. The graphs of these solutions are called solution curves, several of which shown in the figure. Note that both y and yꞌ are defined on (- ) thus for each value of C we have a solution defined for all t. Not all expressions for a general solution include all possible solution to the DE. The general solution to DE yꞌ = y 1/3 is y = [2/3(t + C)] 3/2. But y = 0 is yet another solution which cannot be found by choosing values for C.
Initial Value Problems (IVPs): General solutions involve arbitrary constant C which implies that the ODE has infinitely many solutions. In applications, it is necessary to use other information, in addition to the differential equation, to determine the value of the constant C and to specify the solution completely. Such a solution is called a particular solution. Definition: A first-order differential equation together with an initial condition, yꞌ = f (t, y), y(t 0 ) = y 0, is called an initial value problem. A solution of the initial value problem is a differentiable function y(t) such that 1. yꞌ(t) = f (t, y(t)) for all t in an interval containing t 0 where y(t) is defined, and 2. y(t 0 ) = y 0. this implies point (t 0, y 0 ) is on the particular solution curve
Example: The general solution of DE yꞌ = (y + 1) 1/2 is solution that satisfies y(0) = 2. 2 (C + t) y= -1 4 Set t = 0 and y = 2 in the expression for the general solution. We get to solve to C. The result is C=2 3. Thus the particular solution is 2 (2 3 + t) y= -1 4 Are there any solutions not included in the general solution? Do any of those solutions satisfy the initial condition y(0) = 2?. Find the particular 2 C 2= -1 4
Interval of Existence: The interval of existence of a solution to a differential equation is defined to be the largest interval over which the solution can be defined and remain a solution. It is important to remember that solutions to differential equations are required to be differentiable, and this implies that they are continuous. Example: Find the interval of existence for the solution to the initial value problem The particular solution of the IVP is 2 y' = y, y(0) = 1-1 y(t) = t-1 (verify). The graph of y is a hyperbola with two branches. The function y has an infinite discontinuity at t = 1. Consequently, this function cannot be considered to be a solution to the differential equation y = y 2 over the whole real line. Note that the left branch of the hyperbola in the figure passes through the point (0, 1), as required by the initial condition y(0) = 1. Hence, the left branch of the hyperbola is the solution curve needed. This particular solution curve extends indefinitely to the left, but rises to positive infinity as it approaches the vertical asymptote t = 1 from the left. Thus the interval of existence is (-, 1).
The geometric meaning of a differential equation and its solution.
The direction field is the geometric interpretation of a differential equation. However, the direction field view also gives us a new interpretation of a solution. Associated to the solution y(t), we have the solution curve in the ty-plane. At each point (t, y(t)) on the solution curve the curve must have slope equal to y(t) =f (t, y(t)). In other words, the solution curve must be tangent to the direction field at every point. Thus finding a solution to the differential equation is equivalent to the geometric problem of finding a curve in the ty-plane that is tangent to the direction field at every point. Note how the particular solution curve of IVP y = y, y(0) = 1 is tangent to the direction field at each point (t, y) on the solution curve.
Approximate numerical solutions. The direction field hints at how we might produce a numerical solution of an initial value problem. To find a solution curve for the initial value problem y = f (t, y), y(t 0 ) = y 0, first plot the point P 0 = (t 0, y 0 ). Because the slope of the solution curve at P 0 is given by f (t 0, y 0 ), move a prescribed distance along a line with slope f (t 0, y 0 ) to the point P 1 = (t 1, y 1 ). Next, because the slope of the solution curve at P 1 is given by f (t 1, y 1 ), move along a line with slope f (t 1, y 1 ) to the point P 2 =(t 2, y 2 ). Continue in this manner to produce an approximate solution curve of the initial value problem. The technique described above was used to produce an approximate solution of equation To the IVP y = y, y(0) = 1 and is the basic idea behind Euler s method, an algorithm used to find numerical solutions of initial value problems. Clearly, if we decrease the distance between consecutively plotted points, we should obtain an even better approximation of the actual solution curve.
Using a numeric solver. You need access to a computer. We will have access to a solver that will draw direction fields, provide numerical solutions of differential equations and systems of differential equations, and plot solutions of differential equations and systems of differential equations.
One such solver is dfield8 in MATLAB. Here is a brief description. dfield8 is an interactive tool for studying single first order differential equations. When dfield8 is executed, a dfield8 Setup window is opened. The user may enter the differential equation and specify a display window using the interactive controls in the Setup window. Click to get direction field
When the Proceed button is pressed on the Setup window, the DF Display window is opened. At first this window displays a direction line field for the differential equation. When the mouse button is depressed in the dfield8 Display window, the solution to the differential equation with that initial condition is calculated and plotted. Click your mouse to select an initial value and the graph of the solution to that IVP is shown. Initial condition used was x(2) 1. There is an option to enter the initial condition via the keyboard.
Multiple initial conditions can be selected.
Qualitative methods. So far we do not have techniques to find analytic, closed form solutions. (That is, formulas for general solutions or solutions of IVPs.) However, the lack of closed-form solutions does not prevent us from using a bit of qualitative mathematical reasoning to investigate a number of important qualities of the solutions of this equation. Example: For DE y' = 1 y 2, the direction field provides information about several solutions. Notice that the lines y = 1 and y = 1 seem to be tangent to the direction field. It is easy to verify directly that the constant functions y 1 (t) = -1 and y 2 (t) = 1 are solutions. Each of these is called an equilibrium solution.
From the direction field we can observe the behavior of solutions of either side of the equilibrium solutions. For initial conditions here lim yt ( ) = 1 t For initial conditions here lim yt ( ) = 1 t For initial conditions here lim yt ( ) = 1 t For initial conditions here lim yt ( ) = 1 t