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The proofs of most of the major results are either exercises or Engineering Calculus Homework 2 Homework 2 7. Engineering Mathematics. Firstly, derivation of the basic fluid mechanics equations requires calculus.
Some with solutions. Comment on Iteration Methods Iteration methods were used rather early in history, but it was Picard who made them popular.
Proofs of the theorems in this section given in books of higher level, e. Iterations are well suited for the computer because of their modest storage demand and usually short programs in which the same loop or loops are used many times, with different data. This last expression is the general solution. This does not contradict the theorems because we first have to write the ODE in standard form: For k 0 we still get no solution, violating the existence as in Prob. Thus in this case the uniqueness is violated.
Neither of the two theorems is violated in either case. From this we can see the answers: A unique solution if y x0 equals any y0 and x0 0 or x0 4. The conjecture is true for any choice of a constant or even of a continuous function of x. It was mentioned in footnote 9 that Picard used his iteration for proving his existence and uniqueness theorems. Since the integrations involved in the method can be handled on the computer quite efficiently, the method has gained in importance in numerics.
Formula 4 in Sec. Separate variables.
Use 4 , Sec. Differentiate this with respect to y and equate the result to the coefficient function of dy: Solvable A as a Bernoulli equation or B by separating variables.
This is a linear ODE with unknown u. Solve it by 4 in Sec. Integrate and take exponents on both sides: We now take exponents and simplify algebraically. Logistic equation. Theorem 1 in Sec. We differentiate this with respect to x and equate the result to the coefficient of dx in the exact ODE. In problems of this sort we need two conditions, because we must determine the arbitrary constant c in the general solution and the constant k in the exponent.
We can now determine the time when T reaches This will give a general formula for determining the half-life H from two measurements y1 and y2 at times t1 and t2, respectively. Accordingly, we use letters and insert the given numeric data only at the end of the derivation. Thus the half-life of the substance is about 12 days and 1 hour.
Let y denote the amount of fresh air measured in cubic feet. We use separation of variables. To evaluate the integral, we apply reduction by partial fractions.
We now take exponents. Let the tangent of such a curve y x at x, y intersect the x-axis at M and the y-axis at N, as shown in the figure. This is a family of hyperbolas.
Section 1. For these reasons we consider linear ODEs of third and higher order in a separate chapter, Chap. The new Sec. In the last edition the complex case was discussed in a separate section.
Modeling applications of the method of undetermined coefficients Sec. This avoids discussing the physical aspects and foundations twice. To extend the basic concepts from first-order to second-order ODEs and to present the basic properties of linear ODEs.
Comment on Footnote 1 In , Lagrange gave the first methodical treatment of the calculus of variations. The book mentioned in the footnote includes all major contributions of others in the field and made him the founder of analytical mechanics.
Comment on Terminology p and q are called the coefficients of 1 and 2. The function r on the right is not called a coefficient, to avoid the misunderstanding that r must be constant when we talk about an ODE with constant coefficients. General solution: Section 2. Problem 6 8. Yes when n 2. The intervals given in Probs. Linear independence is important in connection with general solutions, and these problems are such that the computer is of no great help.
The functions are selected as they will occur in some of the later work. They also encourage the student to think of functional relations between those functions. Consider the quotient. Once and for all, we have linear dependence of two or more functions if one of them is identically 0. This problem is important. The only difficulty may be an unpleasant additional integration. The integral of U is tan x; we need no constants of integration because we merely want to obtain a particular solution.
To show that homogeneous linear ODEs with constant coefficients can be solved by algebra, namely, by solving the quadratic characteristics equation 3. The roots may be: To help poorer students, we have shifted the derivation of the real form of the solutions in Case III to the end of the section, but the verification of these real solutions is done immediately when they are introduced.
This will also help to a better understanding. Also it should be emphasized that in the transition from the complex to the real form of the solutions we use the superposition principle. Furthermore, one should emphasize the general importance of the Euler formula 11 , which we shall use on various occasions. Comment on How to Avoid Working in Complex The average engineering student will profit from working a little with complex numbers. Of course, the present approach can be used to handle all three cases.
Team Project. By the superposition principle, another solution is. Differential Operators. Optional, page 59 Purpose.
To take a short look at the operational calculus of second-order differential operators with constant coefficients, which parallels and confirms our discussion of ODEs with constant coefficients. The two conditions follow trivially from the condition in the text. These are free motions no driving force. Forced motions follow in Sec. This system should be regarded as a basic building block of more complicated systems, a prototype of a vibrating system that shows the essential features of more sophisticated systems as they occur in various forms and for various purposes in engineering.
The quantitative agreement between experiments of the physical system and its mathematical model is surprising. Indeed, the student should not miss performing experiments if there is an opportunity, as I had as a student of Prof. Blaess, the inventor of a now obscure graphical method for solving ODEs. Let F be some force that stretches the combination of springs by an amount s0.
The numeric value of the last expression is 1. The dark blue portion of the water in Fig. Then we can choose other values of c by experimentation. In Fig. One is generally surprised how quickly certain states are reached whereas the theoretical time is infinite. The graph shows a positive maximum near 15, a negative minimum near 23, a positive maximum near 30, and another negative minimum at The experiments with the curves are as before in this project.
Euler—Cauchy Equations, page 69 Purpose. Algebraic solution of the Euler—Cauchy equation, which appears in certain applications see our Example 4 and which we shall need again in Sec. We have three cases; this is similar to the situation for constant-coefficient equations, to which the Euler—Cauchy equation can be transformed Team Project 16 ; however, this fact is of theoretical rather than of practical interest. Comment on Footnote 4 Euler worked in St.
Petersburg — and — and in Berlin — His main influence on the development of mathematics and mathematical physics resulted from his textbooks, in particular from his famous Introductio in analysin infinitorum , in which he also introduced many of the modern notations for trigonometric functions, etc. Euler was the central figure of the mathematical activity of the 18th century.
His Collected Works are still incomplete, although some seventy volumes have already been published. Cauchy worked in Paris, except during —, when he was in Turin and Prague.
Cauchy also was the first to give existence proofs in ODEs. He initiated complex analysis; we discuss his main contributions to this field in Secs. His famous integral theorem Sec. A The student should realize that the present steps are the same as in the general derivation of the method in Sec.
An advantage of such specific derivations may be that the student gets a somewhat better understanding of the method and feels more comfortable with it. Of course, once a general formula is available, there is no objection to applying it to specific cases, but often a direct derivation may be simpler.
In that respect the present situation resembles, for instance, that of the integral solution formula for first-order linear ODEs in Sec. This is the expected result. C This is less work than perhaps expected, an exercise in the technique of differentiation also necessary in other cases.
Existence and Uniqueness of Solutions. Wronskian, page 73 Purpose. Of course, no such theory was needed in the last sections on ODEs for which we were able to write all solutions explicitly.
Main Content Continuity of coefficients suffices for existence and uniqueness. Linear independence if and only if the Wronskian is not zero A general solution exists and includes all solutions.