No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the United States States of 1976. Copyright Act, without the prior written consent of the publisher, or authorization by payment of the applicable per copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923 (website: www.copyright.com ). Standalone presentation, except for a few clearly marked places where a proof would exceed the level of the book and a reference is given instead.
GUIDES AND MANUALS
PART A
PART B
PART C
PART D
PART E
PART F
PART G
The variety is endless, but the underlying mathematics is surprisingly powerful and capable of providing advice that leads to the achievement of goals for the betterment of society, for example by recommending wise policies regarding global warming, better allocation of resources in a manufacturing process, or make statistical decisions (such as in section 25.4 about whether a drug is effective in treating a disease). Modeling PDEs is more difficult, so we separated the modeling process from the solution process (in Section 12.6).
Linear Algebra: Matrices, Vectors, Determinants
Complex Numbers and Functions
Basic Concepts. Modeling
Solve ODEs by integration or by memorizing the differentiation formula. a) Check whether y is a solution of the ODE. This is the time during which half of a given amount of radioactive material will disappear.
Geometric Meaning of
First Euler step, showing a solution curve, its tangent at ( ), step hand increases hf(x0, y0) in the formula for y1. In the field graph several solution curves by hand, especially those passing through the given points.
Separable ODEs. Modeling
The volume of the outflow in a short time is. must correspond to the change in the volume of the water in the tank. B Cross-sectional area of tank) where is the drop in water height.
Exact ODEs. Integrating Factors
Experiments show that the change of the force S in a small part of the rope is proportional to Sand in relation to the small angle in fig. You will see that the product of the right sides of the ODEs in (a) and (c) is equal to ⫺ 1. Do you know.
Linear ODEs. Bernoulli Equation
Write (3) as where is the exponential function, which is the solution of the homogeneous linear ODE. A Clairaut equation of the form (15). a) Apply the transformation to the Riccati equation (14), where Y is a solution of (14) and get for the linear ODE.
Orthogonal Trajectories. Optional
Represent the given family of curves in the form and sketch some of the curves. If with fine dependence on y, show that the curves of the corresponding family are congruent, and so are their OTs.
Existence and Uniqueness of Solutions for Initial Value Problems
The electric forces of two opposite charges of equal strength at and are circles through and. Under what conditions does an initial value problem of the form (1) have at least one solution (that is, one or more solutions). Using his method, Picard proved Theorems 1 and 2 and the convergence of the sequence (7) to the solution (1).
This is remarkable because it means that for a linear ODE continuity guarantees not only the existence but also the uniqueness of the solution to an initial value problem. Can two solution curves of the same ODE have a common point in a rectangle in which the assumptions of the present theorems are satisfied.
Homogeneous Linear ODEs of Second Order
1, that for a first-order ODE, an initial value problem consists of the ODE and an initial condition. These conditions prescribe given values and of the solution and its first derivative (the slope of its curve) at the same given in the considered open interval. General solution. The functions and are solutions of the ODE (by example 1), and we take
More in the next set.) (a) Verify that the given functions are linearly independent and form a basis of solutions of the given ODE. Try it out on some of the problems in this and the next problem set and on examples of your own.
Homogeneous Linear ODEs with Constant Coefficients
A general second order ODE, linear or not, can be reduced to first order if y does not occur explicitly (Probably it can be shown that the curve of an inextensible flexible homogeneous cable hanging between two fixed points is obtained by solving y(x) ) ys⫽2yr. If the sum of velocity and acceleration in the movement of a small body in a straight line is equal to a positive constant, how will the distance depend on the initial velocity and position.
Now from algebra we remember that the roots of this quadratic equation are (3). 3) and (4) will be basic because our derivation shows that the functions. From algebra we also know that the quadratic equation (3) can have three types of roots, depending on the sign of the discriminant a24b, namely,. Case I) Two real roots if , (Case II) A real double root if , (Case III) Complex conjugate roots ifa24b0.
Two Distinct Real-Roots and
We further know from algebra that the quadratic equation (3) can have three types of roots, depending on the sign of the discriminant a24b, namely.
Real Double Root
So in the case of a double root of (3) there is a basis of solutions of (1) on each interval.
Complex Roots
Differential Operators. Optional
Therefore the differential calculus involves an operator, the differential operator D, which transforms a (differentiable) function into its derivative. For a homogeneous linear ODE with constant coefficients we can now introduce the second-order differential operator. The extension of operator methods to variable-coefficient ODEs is more difficult and will not be considered here.
If operational methods were limited to the simple situations illustrated in this section, it might not be worth mentioning. In fact, the power of the operator approach appears in more complicated technical problems, as we shall see in Chap.
Modeling of Free Oscillations of a Mass–Spring System
Note that an additional force is present in the spring, caused by its extension in fixing the ball, but has no effect on the motion because it is in equilibrium with the weight W of the ball, , where. The motion of our mass-source system is determined by Newton's second law (2). where and "Force" is the result of all the forces acting on the ball. But if the damping is small and the motion of the system is considered for a relatively short time, we can ignore the damping.
If a mass-spring system with an iron ball of weight nt (about 22 lb) can be considered as inelastic and the spring is such that the ball stretches it 1.09 m (about 43 in.), how many cycles per minute will the system execute. What will be its motion if the ball is pulled down from rest 16 cm (about 6 in.) and allowed to start with zero initial velocity.
Overdamping
Critical Damping
Underdamping
Euler–Cauchy Equations
Real different roots give two real solutions
A real double root occurs if and only if because then (2) becomes as can be easily verified. A second linearly independent solution can be obtained by the method of reduction of order from Sec. Complex conjugate roots are of little practical interest, and we discuss the derivation of real solutions from complex ones only in terms of a typical example.
Complex conjugate roots are of minor practical importance, and we discuss the derivation of real solutions from complex ones just in terms of a typical example
Existence and Uniqueness of Solutions. Wronskian
It concerns the existence and form of a general solution of (1) as well as the uniqueness of the solution of initial value problems consisting of such an ODE and two initial conditions. The two main results will be Theorem 1, which says that such an initial value problem always has a solution that is unique, and Theorem 4, which says that a general solution (3). P R O O F (a) Let be linearly dependent on I. b) Conversely, let for some and show that this implies linear dependence on of I.
If ODE (1) has continuous coefficients p(x) and q(x) in an open interval I, then every solution of (1) in I is of the form. Assume that the coefficients p and q of ODE (1) are continuous on an open interval I, to which the following statements refer. a) Solve (a) with exponential functions, (b) with hyperbolic functions.
Nonhomogeneous ODEs
A general solution of the inhomogeneous ODE (1) on an open interval I is a solution of the form If a term in your choice happens to be a solution of the homogeneous ODE corresponding to (4), multiply this term by x (or by if this solution corresponds to a double root of. Solving the inhomogeneous ODE. The function on the right normally would require the choice.
But we see from this that this function is a solution of the homogeneous ODE, which corresponds to a double root of the characteristic equation. Solving the initial value problem. Setting yand using the first initial condition, we obtain Differentiation of y gives.
Modeling: Forced Oscillations. Resonance
Mechanically, this means that at any moment the resultant of the internal forces is in equilibrium with The resulting movement is called a forced movement with a driving function which is also known as the input driving force and the solution to be obtained is called the output response of the system to the driving force. Of particular interest are external periodic forces, and we will consider a driving force of the form. The cosine terms on both sides must be equal, and the coefficient of the sine term on the left must be zero since there is no sine term on the right.
To eliminate b, multiply the first equation by and the second by and add the results, obtain. We will now discuss the behavior of the mechanical system, distinguishing between the two cases (no damping) and (damping).
Undamped Forced Oscillations. Resonance
Similarly, to eliminate a, multiply (the first equation by and the second by and add to get. If the factor is not zero, we can divide by this factor and solve for a and b. We see that due to the factort, the amplitude of the vibration becomes larger and larger.
Therefore, the period of the last sine function is large, and we obtain an oscillation of the type shown in figure. Forced undamped oscillation when the difference between the input and natural frequencies is small ("beats").
Damped Forced Oscillations
- Modeling: Electric Circuits
- Solution by Variation of Parameters
- Homogeneous Linear ODEs
- Homogeneous Linear ODEs with Constant Coefficients
- Nonhomogeneous Linear ODEs
- For Reference
- Systems of ODEs as Models in Engineering Applications
- Basic Theory of Systems of ODEs
- Constant-Coefficient Systems
- Criteria for Critical Points. Stability
- Qualitative Methods for Nonlinear Systems
- Nonhomogeneous Linear Systems of ODEs
- Power Series Method
If the ODE (2) has continuous coefficients on an open interval I, then any solution of (2) on I has the form. Find a general solution of the given ODE (a) by first converting it into a system (b), as given. The general solution has the form , where there is a general solution of (3) and a special solution of (2).
