In the German school system, a gymnasium is an academic high school that prepares students for university studies. The complex case, where the solutions are holomorphic functions, is treated in 8; the evidence follows the pattern established in 6 for the real case.
Complex Linear Systems
Stability and Asymptotic Behavior
Linear Differential Equations
Boundary Value and Eigenvalue Problems
Motion in the Gravitational Field of Two Bodies (Satellite Orbits). The following equations (10) describe the motion of a small body
- Explicit First Order Equations
We interpret a numerical triple of the form (x, y, p) geometrically in the following way: (x, y) gives a point in the plane, and the third component p gives the slope of a line through the point (x, y) (a , with tan a =p, is the angle of inclination of the line; see the figure). This function gives all positive solutions (this also follows from the uniqueness statement in VII).
The differential equation in the coordinate system. is just the special case c = = 0 of the original equation. Write a first-order differential equation for the following families of curves (parameter cE IR).
The inflection point marks the inflection point where the second derivative becomes negative, and thus the point above which the annual population growth rate begins to decline.
Ly = h(x) The Nonhomogeneous Equation
If a solution q5(x) of the Riccati equation is known, all the other solutions can be obtained in the form. What solutions exist on an infinite interval; which is found in R. b) Determine all solutions of the differential equations.
The Generalized Logistic Differential Equation
- Differential Equations for Families of Curves
By the way, this is the smallest positive solution that exists in all of R; cf. Since the solution z(t) = of the homogeneous equation is unbounded and all solutions of the inhomogeneous equation are given by y = + Az, it follows that.
The part of the curve corresponding to values of t eU can be expressed explicitly in the form. A differential equation of the form (2) is called an exact equation in the domain D if (g, h) is a gradient field, i.e. if there exists a function F(x, y) E C' (D) so that.
Integrating Factors (or Euler Multipliers). The differential equa- tion
- Regular and Singular Line Elements. If = 0 and if equation (2) can be rewritten in a neighborhood U C R3 of the point
- Parametric Representation with y' as the Parameter. In the following sections we will discuss some examples of implicit differential equations
- An Existence and Uniqueness Theorem
- Lemma on the Extension of Solutions. Let D C and f e
- A Proof Based on Zorn's Lemma. Theorem I was derived
- An Elementary Proof of the Peano Existence Theorem
- Existence and Uniqueness Theorem. Let f(x, y) be continuous in a domain D C and satisfy a local Lipschitz condition with respect to y
By Theorem I, there exists exactly one solution y(x) of the initial value problem with right-hand side f. This function y is the right-hand extension of the original solution qS mentioned in the conclusion of the theorem. The function v(x) is called the lower solution (or subsolution) and w(x) is called the upper solution (or supersolution) of the initial value problem.
If f(x,y) is continuous on the strip J x R, J= + a], and if is the minimum and (x) maximum integral of the initial value problem (5), then the solution of the initial value problem (5 ) passes through each point of the set. If f is continuous and the solution y of the initial value problem (5) is unique, then the upper and lower solutions can be characterized by the weak inequalities (11) (with instead of <). Let us denote the (unique) solution of the initial value problem. An) By applying Corollary IX twice to the interval [0, n], which lies to the left of the point = n, it follows that the inequalities v w hold in [0, n].
The initial value problem (6) is equivalent to the integral equation. 7') The set of continuous vector-valued functions defined on J with the norm.
Differential Inequalities and Invariance
In the second step one can assume that F satisfies (for the arguments involved) a Lipschitz condition f(x,y) —f(x,z)I —zi, where is the maximum rate. In the figure, the curves S1 = 0 and S2 : =0 are drawn, and the side of the curve where the component is positive or negative is shown. In the first case, the v-population is extinguished; in the second case it is the u-population which disappears.
A solution starting (at any time) in the upper region converges to C, and one starting in the lower region converges to B. A proof (not simple) is given in (d) below. a) Show that the regions E1 and B3 and Q are negatively invariant and that every solution starting in E1 tends to 0 as t —00. b) The diagonal u = v cuts the first quadrant into a lower part Qi and an upper part Shows that the areas (E1 U E4) fl Q,.
Differential Equations in the Sense of Carat héodory
Existence and Uniqueness Theorem. Let D C be open and suppose that the assumptions of XVIII hold on every set of the form S =
Theorem on Differential Inequalities. Let the function f(x, y)
- Initial Value Problems for Equations of Higher Order
Now consider the integral equation for equivalent to (*) and take a = For n —p oo we obtain an integral equation showing that this is a solution of the initial value problem for y' = f(x,y). It is now a matter of routine to show that under the assumptions of Theorem XX, case n = 1, there exists a maximum solution and a minimum solution and that both solutions approximate the boundary of D to the left and to the right. If the inequality < does not hold in J, then there is a first point c > e such.
Show that with the Eudidean norm, estimate (17) holds even under the weaker assumption (y,(f(x,y)) Here (.,.) is the scalar product v.
Transformation to an Equivalent First Order System. Consider the nth order scalar differential equation in explicit form
- Continuous Dependence of Solutions
We choose a frame of reference such that gravity acts in the negative y-axis. The overhead line equation (and the above derivation) remains valid if the density p is variable. The number v = 1/T =w/(2ir) is called the frequency (the number of oscillations per second), w is the circular frequency and r is the amplitude of the oscillation.
The initial value problem. is locally uniquely solvable in each of the following three cases. a) Let the function f(x, y) be continuous in the strip. Find and sketch all solutions to the initial value problem. to the left and to the right).
Differential Equations for Complex-Valued Functions of a Real Variable. We first extend the notion of an initial value problem by allowing
In the above example, the right-hand side of the differential equation is the Holornorph function of the parameter A E C. Let the vector functions y(x), z(x) and the real function p(x) be defined. and differentiable in the interval J : x + a. Thus, (6) contains the previously proved uniqueness result in the case where the right-hand side satisfies the Lipschitz condition.
Let J be a compact interval with E J and let the function y = yo(x) be a solution to the initial value interval with E J and let the function y = yo(x) be a solution to the initial value problem. In this section, the problem of dependence of the solution of an initial value problem on the data is further explored.
Volterra Integral Equations of the form
We first consider the case where the right-hand side of the differential equation depends on the parameter A; that is f =f(x,y; A). Then there exists a > 0 s Sa C D such that the function y(x; defined in J x Sa (ie, every solution of the initial value problem s 17) Scx . exists at least in J), functions y, and their derivatives with respect to x , denoted by y', are continuous in J x and. In this proof, we can take any element for A0,. and the holomorphism of the solution in Ua follows. a) Note that all derivatives of y with respect to and A' satisfy a linear integral equation of the form
Let y(x; the . ertended solution) of the initial value problem (11) denote and E C lRn+2 the domain of this function (that is, E is the set of all (x; such that the solution of (11) exists from to x) If the functions A(t), B(t), x(t) are differentiable (at a point t0), then each of the following functions is differentiable, and corre-.
Complex Systems Versus Real Systems. Let
- Homogeneous Linear Systems
Show that the norm given in VIII is the operator norm (for all A) and that the two matrix norms defined in II are not operator norms in general (compute the operator norm of A = I). E are linearly independent in if and only if they are linearly independent when viewed as vectors in A similar statement holds for real solutions of (1) when A(t) is real. e) There exist n linearly independent solutions Yi, .., Each such set of n linearly independent solutions is called a fundamental system of solutions. If Yi, .., is a fundamental system, then every solution y can be uniquely written as a linear combination. f) A system of n solutions Yi, .., can be assembled into an n x n solution matrix.
This equation is identical to (2). g) A special fundamental matrix X(t) is obtained from the initial value problem. Using the solution matrix X(t), the solution to any initial value problem can be immediately given:.
Z'=Y'C=AYC=AZ
D'Alembert's Method of Reduction of Order. In general, it is not possible to give the solutions of a homogeneous system in closed form
However, if one solution is known, it is possible to reduce the system to a system of n — 1 differential equations. If x(t) is a (known) solution of the differential equation (1), then one makes the ansatz for the remaining solutions. Here it is assumed, without loss of generality, that x1 (t) 0 (instead of the first component any other component can be chosen).
If (11) is completely solved, that is, if a fundamental system has been found, this procedure leads to n — 1solutions Yi, .., Ym1 of the original differential equation (1). Since the first component of each z2 vanishes, the first component of this equation is divided by x1.
Example. The system
- Inhomogeneous Systems
As in the case n = 1, the proof rests on the simple fact that the difference between two solutions of the inhomogeneous differential equation is a solution. The first sum on the right is a solution of the homogeneous equation with the initial value rj, and the second sum is a solution of the inhomogeneous equation with initial value 0 (see (3)). Thus a representation of the solution y to the initial value problem in terms of Y(t) is given by.
Our next theorem concerns the linear case of Equation Theorem 10.XII in the context of C-solutions. A similar result also holds if some of the A2 are equal, that is, if there are multiple eigenvalues.
Example
The independence of these real solutions follows from the fact that the original solutions = (i = 1, .. 2p + q) are linearly independent of Theorem II and can be represented as linear combinations of the above real solutions; cf. Thus, if B C—1AC has Jordan normal form, then each column of Z(t) is a solution of (7) of the form. Systems of Constant Coefficients 183 The degree of the polynomials that arise can be determined from the Jordan normal form.
If A is a k-fold zero of the characteristic polynomial of A, then the number m(A) := kis is called the algebraic multiple of the eigenvalue, and the dimension m' (A) of the corresponding eigenspace, that is say the maximum number of its linearly independent eigenvectors, is called the geometric multiplicity. The calculation of the solutions is easily accomplished once the Jordaan normal form B =C'AC and the transformation matrix C have been determined.
