VERTEBRACIÓN DEL TERRITORIO, MOVILIDAD Y VIVIENDA
ANEXO – RESUMEN CREDITOS PROPUESTOS ESTIMADOS
6.1 Historical introduction
The history of differential equations began in the 17th century when Newton, Leibniz, and the Bernoullis solved some simple differential equations of the first and second order arising from problems in geometry and mechanics. These early discoveries, beginning about 1690, seemed to suggest that the solutions of all differential equations based on geometric and physical problems could be expressed in terms of the familiar elementary functions of calculus. Therefore, much of the early work was aimed at developing ingenious techniques for solving differential equations by elementary means, that is to say, by addition, sub-traction, multiplication, division, composition, and integration, applied only a finite number of times to the familiar functions of calculus.
Special methods such as separation of variables and the use of integrating factors were devised more or less haphazardly before the end of the 17th century. During the 18th century, more systematic procedures were developed, primarily by Euler, Lagrange, and It soon became apparent that relatively few differential equations could be solved by elementary means. Little by little, mathematicians began to realize that it was hopeless to try to discover methods for solving all differential equations. Instead, they found it more fruitful to ask whether or not a given differential equation has any solution at all and, when it has, to try to deduce properties of the solution from the differential equation itself.
Within this framework, mathematicians began to think of differential equations as new sources of functions.
An important phase in the theory developed early in the 19th century, paralleling the general trend toward a more rigorous approach to the calculus. In the
obtained the first “existence theorem” for differential equations. He proved that every first-order equation of the form
Y’ =
has a solution whenever the right member, satisfies certain general conditions.
One important example is the Ricatti equation
= + +
where Q, and are given functions. Cauchy’s work implies the existence of a solution of the Ricatti equation in any open interval r) about the origin, provided
P, Q,
and 1 4 2Review of results concerning linear equations of first and second orders 143 R have power-series expansions in r). In 1841 Joseph Liouville (1809-1882) showed that in some cases this solution cannot be obtained by elementary means.
Experience has shown that it is difficult to obtain results of much generality about solutions of differential equations, except for a few types. Among these are the so-called linear differential equations which occur in a great variety of scientific problems. Some simple types were discussed in Volume I-linear equations of first order and linear equations of second order with constant coefficients. The next section gives a review of the principal results concerning these equations.
6.2 Review of results concerning linear equations of first and second orders A linear differential equation of first order is one of the form
Y
’ + =
where P and Q are given functions. In Volume I we proved an existence-uniqueness theorem for this equation (Theorem 8.3) which we restate here.
THEOREM 6.1. Assume P and Q are continuous on an open interval Choose any point a in J and let b be any real number. Then there is one and one function y = (x) which satisfies the equation (6.1) and the initial condition f(a) = b . This function is given by the explicit formula
+ dt ,
where A(x) = P(t) dt .
Linear equations of second order are those of the form
+ + =
If the coefficients and the right-hand member R are continuous on some interval J, and if is never zero on J, an existence theorem (discussed in Section 6.5) guarantees that solutions always exist over the interval J. Nevertheless, there is no general formula analogous to (6.2) for expressing these solutions in terms of , and R. Thus, even in this relatively simple generalization of the theory is far from complete, except in special cases. If the coefficients are constants and if R is zero, all the solutions can be determined explicitly in terms of polynomials, exponential and trigonometric functions by the following theorem which was proved in Volume I (Theorem 8.7).
THEOREM 6.2. Consider the equation
144 Linear differential equations
where a and b are given real constants. Let d = 4b. Then every solution of (6.3) on the interval + has the form
(6.4) y
= +
where and are constants, and the functions and are determined according to the algebraic sign of d as
(a) d then = 1 and = x .
(b) d > then = and = where k = (c) If d < then = cos kx and = sin kx, where k =
The number d = 4b is the discriminant of the quadratic equation
This is called the characteristic equation of the differential equation (6.3). Its roots are given
- a + - a
= , .
The algebraic sign of d determines the nature of these roots. If d > 0 both roots are real and the solution in (6.4) can be expressed in the form
y = +
If d < the roots and are conjugate complex numbers. Each of the complex exponential functions = e and = is a complex solution of the differential equation (6.3). We obtain real solutions by examining the real and imaginary parts and Writing = + ik , = ik, where k = we have
+
and
The general solution appearing in Equation (6.4) is a linear combination of the real and imaginary parts of and
6.3 Exercises
These exercises have been selected from Chapter 8 in Volume I and are intended as a review of the introductory material on linear differential equations of first and second orders.
Linear equations order. In Exercises solve the initial-value problem on the specified interval.
= O w h e n x = O .
Linear equations of order 145 4. If a strain of bacteria grows at a rate proportional to the amount present and if the population
doubles in one hour, by how much will it increase at the end of two hours?
5. A curve with Cartesian equation passes through the origin. Lines drawn parallel to the coordinate axes through an arbitary point of the curve form a rectangle with two sides on the axes. The curve divides every such rectangle into two regions A and B, one of which has an area equal to times the other. Find the
6. (a) Let be a solution of the second-order equation y” + + = 0.
Show that the substitution y = converts the equation
=
into a first-order linear equation for .
(b) Obtain a solution of the equation + = 0 by inspection and use the method of part (a) to find a solution of
+ (y’ =
Linear equations order with constant In each of Exercises 7 through 10, find all solutions on ( +
7. - 4 y = o . 9.
8. y” + = 0. 10.
11. Find all values of the constant k such that the differential equation y” + ky = 0 has a nontrivial solution y for which&(O) = 0. For each permissible value of k, determine the corresponding solution y . Consider both positive and negative values of k.
12. If (a, is a given point in the plane and if is a given real number, prove that the differential equation + = 0 has exactly one solution whose graph passes through (a, and has slope m there. Discuss separately the case k = 0.
13. In each case, find a linear differential equation of second order satisfied by and
= , =
(b) = =
(c) = cos x, = sin x.
(d) = sin (2x + = sin (2x + 2).
(e) = x, sinh x .
14. A particle undergoes simple harmonic motion. Initially its displacement is 1, its velocity is 1 and its acceleration is 12. Compute its displacement and acceleration when the velocity is
6.4 Linear differential equations of order
A linear differential equation of order n is one of the form
+ + + = R(x)
The functions , , . . . , multiplying the various derivatives of the unknown function y are called the of the equation. In our general discussion the linear equation we shall assume that all the coefficients are continuous on some interval The word
“interval” will refer either to a bounded or to an unbounded interval.
146 Linear equations
In the differential equation (6.6) the leading coefficient plays a special role, since it determines the order of the equation. Points at which P,(x) = 0 are called singular points of the equation. The presence of singular points sometimes introduces complications that require special investigation. To avoid these difficulties we assume that the function is never zero on J. Then we can divide both sides of Equation (6.6) by and rewrite the differential equation in a form with leading coefficient 1. Therefore, in our general dis-cussion we shall assume that the differential equation has the form
(6.7) + + + = R(x).
The discussion of linear equations can be simplified by the use of operator notation. Let denote the linear space consisting of all real-valued functions continuous on an interval J. Let denote the consisting of all functions
f
whose first derivatives . exist and are continuous on J. Let . . . , be given functions in and consider the operator L: V(J) given byL(f) f + + + The operator L itself is sometimes written as
where denotes the kth derivative operator. In operator notation the differential equation in (6.7) is written simply as
L(y) = R.
A solution of this equation is any function in which satisfies (6.8) on the interval It is easy to verify that + = + and that = for every constant c. Therefore L is a linear operator. This is why the equation L(y) R is referred to as a linear equation. The operator L is called a linear operator of order n.
With each linear equation L(y) R we may associate the equation
=
in which the right-hand side has been replaced by zero. This is called the homogeneous equation corresponding to L(y) = R. When R is not identically zero, the equation = R is called a nonhomogeneous equation. We shall find that we can always solve the non-homogeneous equation whenever we can solve the corresponding non-homogeneous equation.
Therefore, we begin our study with the homogeneous case.
The set of solutions of the homogeneous equation is the null space N(L) of the operator L. This is also called the solution space of the equation. The solution space is a
of Although . infinite-dimensional, it turns out that the solution space N(L) is always finite-dimensional. In fact, we shall prove that
(6.9) dim N(L) =
The dimension of the solution space of a homogeneous linear equation 147 where is the order of the operator L. Equation (6.9) is called the dimensionality theorem for linear differential operators. The dimensionality theorem will be deduced as a conse-quence of an existence-uniqueness theorem which we discuss next.
6.5 The existence-uniqueness theorem
THEOREM 6.3. FOR ORDER Let
. , be continuous functions on an open interval J, let L be the linear operator
are given real numbers, then there exists one and only one function y = f(x) which satisfies the homogeneous equation L(y) = on J and
which also satisfies the initial conditions
= = . . . = .
Note: The vector in n-space given by . . . is called the initial-value vector off at Theorem 6.3 tells us that if we choose a point in and choose a vector in n-space, then the homogeneous equation = 0 has exactly one solution =
f
(x) on J with this vector as initial-value vector at For example, when= 2 there is exactly one solution with prescribed value and prescribed derivative at a prescribed point
The proof of the existence-uniqueness theorem will be obtained as a corollary of more general existence-uniqueness theorems discussed in Chapter 7. An alternate proof for the case of equations with constant coefficients is given in Section 7.9.
6.6 The dimension of the solution space of a homogeneous linear equation
THEOREM 6.4. DIMENSIONALITY THEOREM. Let be a linear operator of order given by
(6.10) +
Then the solution space of the equation L(y) = 0 has dimension n.
Proof. Let denote the n-dimensional linear space of n-tuples of scalars. Let T be the linear transformation that maps each function in the solution space N(L) onto the value vector off at
=
. . .where is a fixed point in J. The uniqueness theorem tells us that T(f) = 0 implies = 0.
Therefore, by Theorem 2.10, T is one-to-one on N(L). Hence is also one-to-one and maps onto N(L)!. and Theorem 2.11 shows that dim N(L) = dim = .
148 Linear equations
Now that we know that the solution space has dimension n, any set of n independent solutions will serve as a Therefore, as a corollary of the dimensionality theorem we have :
THEOREM 6.5. Let L: V(J) be a linear operator of order n. If . . . , are independent solutions of the homogeneous differential equation L(y) = 0 on J, then every solution y = (x) on J can be expressed in the form
(6.11)
where . . . , are constants.
Note: Since all solutions of the differential equation L(y) = 0 are contained in formula (6.1 I), the linear combination on the right, with arbitrary constants . . . , c,, is sometimes called the general solution of the differential equation.
The dimensionality theorem tells us that the solution space of a homogeneous linear differential equation of order always has a basis of n solutions, but it does not tell us how to determine such a basis. In fact, no simple method is known for determining a basis of solutions for every linear equation. However, special methods have been devised for special equations. Among these are differential equations with constant coefficients to which we turn now.
6.7 The algebra of constant-coefficient operators
A constant-coefficient operator A is a linear operator of the form
(6.12) + + + + a,,
where D is the derivative operator and a,, a,, . . . , a, are real constants. If a, 0 the operator is said to have order n. The operator A can be applied to any function y with n derivatives on some interval, the result being a function A(y) given by
A(y) + + + +
In this section, we restrict our attention to functions having derivatives of every order on co, + co). The set of all such functions will be denoted by and will be referred to as the class of functions. If y then A(y) is also in
The usual algebraic operations on linear transformations (addition, by scalars, and composition or multiplication) can be applied, in particular, to constant-coefficient operators. Let A and B be two constant-constant-coefficient operators (not necessarily of the same order). Since the sum A + B and all scalar IA are also constant-coefficient operators, the set of all constant-constant-coefficient operators is a linear space. The product of A and B (in either order) is also a constant-coefficient operator. Therefore, sums, products, and scalar multiples of constant-coefficient operators satisfy the usual commutative, associative, and distributive laws satisfied by all linear transformations.
Also, since we have = for all positive integers r and any two constant-coefficient operators commute; AB = BA .