He is a member of the American Mathematical Society, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. The study of differential equations has attracted the attention of many of the world's greatest mathematicians during the past three centuries.
Some Basic Mathematical Models;
Write the differential equation for the concentration of the chemical in the pond at time t. Hint: Concentration is c=a/v=a(t)/106. Note that the differential equation is the same whether the temperature of the object is above or below the ambient temperature.
Solutions of Some Differential Equations
In Example 1, we found infinitely many solutions of the differential equation (4), which correspond to the infinitely many values that the arbitrary constantsine equation (11) can have. The solution (19) confirms the conclusions reached on the basis of the direction field and Example 1.
Undetermined Coefficients. Here is an alternative way to solve the equation
Classification of Differential Equations
The order of a differential equation is the order of the highest derivative that appears in the equation. In each of Exercises 5 to 10, check that each given function is a solution of the differential equation.
Linear Differential Equations; Method of Integrating Factors
The choice of t0 determines a special value of the constant, but does not change the solution. If g(t) is not zero everywhere, assume that the solution of equation (48) has the form
Separable Differential Equations
Any differentiable function y = φ(x) satisfying equation (13) is a solution of equation (4); in other words, equation (13) defines the solution implicitly rather than explicitly. Note that the boundary of the validity interval of the solution (21) is determined by the point (−2, 1) where the tangent line is vertical.
Modeling with First-Order Differential Equations
Therefore, variations in the amount of salt are solely due to the flows in and out of the tank. Then the solution of the initial value problem (8) gives the balance S(t) in the account at each time t.
Brachistochrone Problem. One of the famous problems in the history of mathematics is the brachistochrone 9 problem: to find
Differences Between Linear and Nonlinear Differential Equations
So far we have discussed a number of initial value problems, each of which had a solution and apparently only one solution. This raises the question of whether this is true of all initial value problems for first-order equations.
Existence and Uniqueness Theorem for First-Order Linear Equations
If you encounter a problem of initial value while investigating a physical problem, you may want to know that it has a solution before spending too much time and effort to find it. Finally, the initial condition (2) determines the constant in an unusual way, so there is only one solution to the initial value problem; this completes the proof.
Existence and Uniqueness Theorem for First-Order Nonlinear Equations
Autonomous Differential Equations and Population Dynamics
Thus, the mathematical model consisting of the initial value problem (1), (2) with > 0 predicts that the population will grow exponentially for all time, as shown in Figure 2.5.1 for some values of y0. We first look for solutions of equation (7) of the simplest possible type---that is, constant functions. In the same way, any equilibrium solution of the more general equation (1) can be found by setting the roots of f(y) =0.
Therefore, the graphs for solutions of equation (7) must have the general form shown in Figure 2.5.3b, regardless of the values of randK. After a long time the population is close to the saturation level K regardless of the initial population size as long as it is positive. Note that this equation (apart from replacing the parameter K by T) differs from the logistic equation (7) only by the presence of the minus sign on the right-hand side.
For example, in fluid mechanics, equations of the form (7) or (14) often govern the development of a small perturbation y in a laminar (or smooth) fluid flow.
Semistable Equilibrium Solutions. Sometimes a constant equilibrium solution has the property that solutions lying on one side
Obtaining a renewable resource. Suppose that the population of a certain type of fish (for example, tuna or halibut) in a given area of the ocean is described by a logistic equation. In this problem, we assume that fish are caught at a constant rate, independent of the size of the fish population. Assume that the disease spreads through contact between sick and healthy members of the population and that the rate of spread d y/d is proportional to the number of such contacts.
Find the proportion of the population that escapes the epidemic by finding the limiting value ofxast→. Problems 24 through 26 describe three types of bifurcations that can occur in simple equations of the form (28). Note: If we plot the location of the critical points as a function of a y-plane, we get Figure 2.5.10.
Draw the bifurcation diagram for equation (30) --- that is, plot the location of the critical points versus .
Chemical Reactions. A second-order chemical reaction involves the interaction (collision) of one molecule of a substance
- Exact Differential Equations and Integrating Factors
- Numerical Approximations
- The Existence and Uniqueness Theorem
A partial differential equation of the form (26) can have more than one solution; if this is the case, such a solution can be used as an integrator of equation (23). Second, it is usually not possible to find the solutionφ by symbolic manipulations of the differential equation. Compare them with the corresponding values of the actual solution of the initial value problem.
The sixth column contains values of the solution (12) of the initial value problem (9), correct to five decimal places. Use Euler's method with various step sizes to calculate approximate values of the solution for 0≤t≤5. The general solution of this differential equation was found in Example 2 of Section 2.1, and the solution of the initial value problem (11) is.
Use Euler's method with various step sizes to find approximate values of the solution on the interval 0≤t≤5.
First-Order Difference Equations
The solution of the differential equation (12) with this value for ρ and the initial condition y0=10,000 is given by equation (15); that is,. Therefore, we conclude that for this range of ρ values the equilibrium solution un =(ρ−1)/ρ is asymptotically stable. For ρ > 3, none of the equilibrium solutions is stable and the solutions of equation (21) show increasing complexity as ρ increases.
In each of Problems 1 through 4, solve the given difference equation in terms of the initial value0. What is the total amount paid during the term of the loan in each of these cases? This illustrates that the long-term behavior of the solution is independent of the initial conditions.
This is an estimate of the value of ρ at which the onset of chaos occurs in the solution of the logistic equation (21).
Riccati Equations. The equation d y
- Homogeneous Differential Equations with Constant Coefficients
- Solutions of Linear Homogeneous Equations; the Wronskian
In the same way, the functionsetc1y1(t)=c1etandc2y2(t)=c2e−ts satisfy the differential equation (9) for all values of the constantvec1andc2. Its importance lies in the fact that ifr is a root of the polynomial equation (17), then y = er t is a solution of the differential equation (8). The general solution of the differential equation was found in example 2 and is given by equation (27).
In this example, the initial slope is 3, but the solution of the given differential equation behaves similarly for any other positive initial slope. Meanwhile, in Section 3.2 we provide a systematic discussion of the mathematical structure of the solutions of all second-order linear homogeneous equations. Then determine the maximum value of the solution and also find the point where the solution is zero.
In the previous section, we showed how to solve some differential equations of the form ay′′+by′+cy=0,.
Existence and Uniqueness Theorem)
Now we build on these results to provide a clearer picture of the structure of the solutions of all second-order linear homogeneous equations. The fundamental theoretical result for initial value problems for second-order linear equations is given in Theorem 3.2.1, which is analogous to Theorem 2.4.1 for first-order linear equations. Nevertheless, Theorem 3.2.1 says that this solution is indeed the only solution to the initial value problem (5).
For most problems of the form (4), it is not possible to write a useful expression for the solution. Therefore, all parts of the theorem must be proved by general methods that do not involve such an expression. By the unique part of Theorem 3.2.1, it is the only solution of the given problem.
2The proof of Theorem 3.2.1 can be found, for example, in Chapter 6, Section 8 of Coddington's book, which is listed in the references at the end of this chapter.
Principle of Superposition)
The unique part of Theorem 3.2.1 guarantees that these two solutions of the same initial value problem are indeed the same function; thus, for the right choice ofc1andc2, . Then (by Theorem 3.2.3) there are values ofy0andy0′ such that no values ofc1andc2 satisfy the system (8). We can reproduce the result of Theorem 3.2.4 in a slightly different language: to find the general solution, and therefore all solutions, of an equation of the form (2), we only need to find two solutions of the given equation if the Wronskian is not zero .
In several cases we have succeeded in finding a fundamental set of solutions, and thus the general solution, of a given differential equation. Also note that Theorem 3.2.5 does not address how to find the solutions y1andy2 by solving the specified initial value problems. However, they are not the fundamental solutions indicated by Theorem 3.2.5, because they do not satisfy the initial conditions stated in that theorem at point t=0.
While this can be proved by an argument similar to the one just used to prove Theorem 3.2.6, it also follows from Theorem 3.2.2 since y=u(t)−i v(t) is a linear combination of the two solutions.
Abel’s Theorem) 4
Exact Equations. The equation
- Complex Roots of the Characteristic Equation
In each of Problems 36 and 37, use the result of Problem 35 to find the adjoint of the given differential equation. There are several ways to discover how this expansion of the exponential function should be defined. The value of an exponential function with a complex exponent is a complex number whose real and imaginary parts are given by terms on the right side of equation (14).
To do this, we can make use of Theorem 3.2.6, which states that the real and imaginary parts of a complex valued solution of equation (16) are also solutions of the same differential equation. Consequently, if the roots of the characteristic equation are complex numbersλ±iμ, withμ =0, then the general solution of equation (1) is Note that if the real part of the roots is zero, as in this example, then there is no exponential factor in the solution.
Note that this result is illustrated by the solutions y1(t) = cost and y2(t) = sint of the equation y′′+y=0.
- Repeated Roots; Reduction of Order
- Nonhomogeneous Equations; Method of Undetermined Coefficients
- Variation of Parameters
- Mechanical and Electrical Vibrations
- Forced Periodic Vibrations
- General Theory of n th Order Linear Differential Equations
- Homogeneous Differential Equations with Constant Coefficients
- The Method of Undetermined Coefficients
- The Method of Variation of Parameters
If Y1andY2 are two solutions of the non-homogeneous linear differential equation (1), then their difference Y1−Y2 is a solution of the corresponding homogeneous differential equation (2). Form the sum of the general solution of the homogeneous equation (step 1) and the particular solution of the non-homogeneous equation (step 5). Use the result of Problem 16 to show that solving the initial value problem.
The position of the mass is (approximately) given by the solution of the linear differential equation of the second order (7), subject to the prescribed initial conditions (8).
