Modeling with differential equations

1. Modeling with differential equations • One of the most important application of calculus is differential equations, which often arise in describing some phenomenon in engineering, physical science and social science as well.

2. Concepts of differential equations • In general, a differential equation is an equation that contains an unknown function and its derivatives. The order of a differential equation is the order of the highest derivative that occurs in the equation. • A function y=f(x) is called a solution of a differential equation if the equation is satisfied when y=f(x) and its derivatives are substituted into the equation.

3. Example • Ex. Show that every member of the family of functions where c is an arbitrary constant, is a solution of • Sol.

4. Concepts of differential equations • If no additional conditions, the solution of a differential equation always contains some constants. The solution family that contains arbitrary constants is called the general solution. • In real applications, some additional conditions are imposed to uniquely determine the solution. The conditions are often taken the form that is, giving the value of the unknown function at the end point. This kind of condition is called an initial condition, and the problem of finding a solution that satisfies the initial condition is called an initial-value problem.

5. Geometric point of view • Geometrically, the general solution is a family of solution curves, which are called integral curves. • When we impose an initial condition, we look at the family of solution curves and pick the one that passes through the point • Physically, this corresponds to measuring the state of a system at time and using the solution of the initial-value problem to predict the future behavior of the system.

6. Example • Ex. Solve the initial-value problem • Sol. Since the general solution is substituting the values t=0 and y=2, we have So the solution of the initial-value problem is

7. Graphical approach: direction fields • For most differential equations, it is impossible to find an explicit formula for the solution. • Suppose we are asked to sketch the graph of the solution of the initial-value problem • The equation tells us that the slope at any point (x,y) on the graph is f(x,y). • To sketch the solution curve, we draw short line segments with slope f(x,y) at a number of points (x,y). The result is called a direction field.

8. Example • Ex. Draw a direction for the equation What can you say about the limiting value when • Sol. • Remark: equilibrium solution

9. Separable equations • Not all equations have an explicit formula for a solution. But some types of equations can be solved explicitly. Among others, separable equations is one type. • A separable equation is a first-order differential equation in which the expression for can be factored into the product of a function of x and a function of y. That is, a separable equation can be written in the form

10. Solutions of separable equations • Thus a separable equation can be written into that is, the variables x and y are separated! • We can then integrate both sides to get: • After we find the indefinite integrals, we get a relationship between x and y, in which there generally has an arbitrary constant. So the relationship determines a function y=y(x) and it is the general solution to the differential equation.

11. Example • Ex. Solve the differential equation • Sol. Rewrite the equation into Integrate both sides which gives So the general solution is

12. Example • Ex. Solve the differential equation • Sol. Separate variables: Integrate: which is the general solution in implicit form. • Remark: it is impossible to solve y in terms of x explicitly.

13. Example • Ex. Solve the differential equation • Sol. C is arbitrary, but is not arbitrary. While we can verify y=0 is also a solution. Therefore where A is an arbitrary constant, is the general solution.

14. Orthogonal trajectories • An orthogonal trajectory of a family of curve is a curve that intersects each curve of the family orthogonally. For instance, each member of the family of straight lines is an orthogonal trajectory of the family • To find orthogonal trajectories of a family of curve, first find the slope at any point on the family of curve, which is generally a differential equation. At any point on the orthogonal trajectories, the slope must be the negative reciprocal of the aforementioned slope. So the slope of orthogonal trajectories is governed by a differential equation, too. Last solve the equation to get the orthogonal trajectories.

15. Example • Ex. Find the orthogonal trajectories of the family of curves where k is an arbitrary constant. • Sol. Differentiating we get or Substituting into it, we find the slope at any point is At any point on orthogonal trajectory, the slope is Solving the equation, we get

16. Example • Ex. Suppose f is continuous and Find f(x). • Sol.

17. Homework 21 • Section 8.2: 8, 14, 29 • Section 8.3: 28, 29 • Page 583: 7, 8, 10 • Section 9.1: 10, 11