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SCHAUM'S. OUTLINE OF. DIFFERENTIAL. EQUATIONS. Third Edition. RICHARD BRONSON, Ph.D. Professor of Mathematics and Computer Science. Fairleigh. and depression to functional syndromes like irritable bowel, fibromyalgia Dummies, is a member of the Association for Schaum's Outline of Discrete. DIFFERENTIAL EQUATIONS Based on Schaum's O u t l i n e o f T h e o r y a n d P ro b l e m s o f D i f f e re n t i a l E q u a t i o n s, S e c o n d E d i t i o n b y R.

Here c1, c2, and w are constants with w often referred to as circular frequency. No notes for slide. We make the substitution suggested by 2. Separable Equations Consider a differential equation in differential form 1. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Alternatively, the solution to 2.

Like this presentation? Why not share! An annual anal Embed Size px. Start on. Show related SlideShares at end. WordPress Shortcode. Published in: Full Name Comment goes here. Are you sure you want to Yes No. Be the first to like this. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Book details Author: Richard Bronson Pages: McGraw-Hill Education Language: English ISBN Description this book Title: Example 7.

A steel ball weighing lb is suspended from a spring, whereupon the spring is stretched 2 ft from its natural length. The applied force responsible for the 2-ft displacement is the weight of the ball, lb. For convenience, we choose the downward direction as the positive direction and take the origin to be the center of gravity of the mass in the equilibrium position. We assume that the mass of the spring is negligible and can be neglected and that air resistance, when present, is proportional to the velocity of the mass.

Thus, at any time t, there are three forces acting on the system: Second-Order Linear Differential Equations 49 proportionality. Note that the restoring force Fs always acts in a direction that will tend to return the system to the equilibrium position: We automatically compensated for this force by measuring distance from the equilibrium position of the spring. If one wishes to exhibit gravity explicitly, then distance must be measured from the bottom end of the natural length of the spring.

Electrical Circuit Problems The simple electrical circuit shown in Figure consists of a resistor R in ohms; a capacitor C in farads; an inductor L in henries; and an electromotive force emf E t in volts, usually a battery or a generator, all connected in series.

The algebraic sum of the voltage drops in a simple closed electric circuit is zero. The second initial condition is obtained from Equation 7.

Buoyancy Problems Consider a body of mass m submerged either partially or totally in a liquid of weight density r. Such a body experiences two forces, a downward force due to gravity and a counter force governed by: A body in liquid experiences a buoyant upward force equal to the weight of the liquid displaced by that body.

Figure Equilibrium occurs when the buoyant force of the displaced liquid equals the force of gravity on the body. Figure depicts the situation for a cylinder of radius r and height H where h units of cylinder height are submerged at equilibrium.

We arbitrarily take the upward direction to be the positive x-direction. If the cylinder is raised out of the water by x t units, as shown in Figure , then it is no longer in equilibrium. For electrical circuit problems, the independent variable x is replaced either by q in Equation 7. For damped motion, there are three separate cases to consider, according as the roots of the associated characteristic equation see Chapter Five are 1 real and distinct, 2 equal, or 3 complex conjugate.

A steady-state motion or current is one that is not transient and does not become unbounded. Free damped systems always yield transient motions, while forced damped systems assuming the external force to be sinusoidal yield both transient and steady-state motions.

Here c1, c2, and w are constants with w often referred to as circular frequency. Equation 7. The solution of 7. Solved Problem 7. Let h denote the length in feet of the submerged portion of the cylinder at equilibrium.

If this limit does not exist, the improper integral diverges and f x has no Laplace transform. When evaluating the integral in Equation 8. The Laplace transforms for a number of elementary functions are given in Appendix A. Properties of Laplace Transforms Property 8. Laplace Transforms and Inverse Transforms Property 8. They are equally applicable for functions of any independent variable and are generated by replacing the variable x in the above equations by any variable of interest.

In particular, the counter part of Equation 8.

The simplest technique for identifying inverse Laplace transforms is to recognize them, either from memory or from a table such as in the Appendix. Observe from the Appendix that almost all Laplace transforms are quotients. Manipulating Denominators The method of completing the square converts a quadratic polynomial into the sum of squares, a form that appears in many of the denominators in the Appendix.

The method is carried out as follows. Finally, solve these equations for Ai, Bj, and Ck.

Property 8. Theorem 8. Convolution Theorem. Laplace Transforms and Inverse Transforms 61 1 Theorem 8. Using Equation 8. See also entry 7 in the Appendix. Solved Problem 8. This problem can be done three ways. Using Property 8. Using the results of Problem 8.

Laplace transforms convert differential equations into algebraic equations. First, take the Laplace transform of both sides of Equation 9. There are two exceptions: Solutions by Laplace Transforms 67 and the solution to differential equation 9. They are then evaluated separately when appropriate subsidiary conditions are provided. Solutions of Linear Systems Laplace transforms are useful for solving systems of linear differential equations; that is, sets of two or more differential equations with an equal number of unknown functions.

Laplace transforms are taken of each differential equation in the system; the transforms of the unknown functions are determined algebraically from the resulting set of simultaneous equations; inverse transforms for the unknown functions are calculated with the help of the Appendix. Solved Problems Solved Problem 9.

Taking the Laplace transform of both sides of this differential equation and using Property 8. Then, using the Appendix and Equation 9. Solved Problem 9. In this book, the elements of matrices will always be numbers or functions of the variable t. If all the elements are numbers, then the matrix is called a constant matrix. Example A matrix is square if it has the same number of rows and columns.

A vector designated by a lowercase boldface letter is a matrix having only one column or one row. The third matrix given in Example Matrix multiplication is associative and distributes over addition; in general, however, it is not commutative. Theorem Cayley-Hamilton theorem.

However, it follows with some effort from Theorem Let A be as in Theorem When computing the various derivatives in Matrices and the Matrix Exponential 75 Example Method of computation: For each eigenvalue li of At, apply Theorem When this is done for each eigenvalue, the set of all equations so obtained can be solved for a0,a1, These values are then substituted into Equation Solved Problems Solved Problem From Equation Substituting these values successively into Equation The method of reduction is as follows.

Step 1. Rewrite Step 2. Equations This last equation is an immediate consequence of Equations Reduction of a System A set of linear differential equations with initial conditions also can be reduced to system The procedure is nearly identical to the method for reducing a single equation to matrix form; only Step 2 changes. The solution to Equation Usually x t is obtained quicker from However, the integrals arising in All constants of integration can be disregarded when computing the integral in Equation For this A, eAt is given in Problem If b2 x is not zero in a given interval, then we can divide by it and rewrite Equation In this chapter, we describe procedures for solving many equations in the form of You Need to Know!

Polynomials, sin x, cos x, and ex are analytic everywhere. Sums, differences, and products of polynomials, sin x, cos x, and ex are also analytic everywhere.

Quotients of any two of these functions are analytic at all points where the denominator is not zero. Power Series Solutions 87 The point x0 is an ordinary point of the differential equation If either of these functions is not analytic at x0, then x0 is a singular point of Step 3.

Solve this equation for the aj term having the largest subscript. The resulting equation is known as the recurrence formula for the given differential equation. Step 4. Step 5. Although the differential equation must be in the form of Equation In Step 1, Equations Initial-Value Problems Important! Then the solution of the original equation is easily gotten by back-substitution. Method of Frobenius Theorem Power Series Solutions 91 are substituted into Equation Terms with like powers of x are collected together and set equal to zero.

When this is done for xn the resulting equation is a recurrence formula. The two roots of the indicial equation can be real or complex. In this book we shall, for simplicity, suppose that both roots of the indicial equation are real. General Solution The method of Frobenius always yields one solution to Equation The general solution see Theorem 4. The method for obtaining this second solution depends on the relationship between the two roots of the indicial equation.

Substitute these an into Equation If it yields a second solution, then this solution is y2 x , having the form of Solved Problem It follows from Problem Substituting Equations Successively evaluating the recurrence formula obtained in Problem Substituting Power Series Solutions 95 are analytic everywhere: Note that for either value of l, Equation Thus, k!

Equation Table The two operations given above are often used in concert. Using Equation Observe that in a particular problem, f x, y may be independent of x, of y, or of x and y. The graphs of solutions to If the left side of Equation To obtain a graphical approximation to the solution curve of Equations Denote the terminal point of this line element as x1, y1. Then construct a second line element at x1, y1 and continue it a short distance.

Denote the terminal point of this second line element as x2, y2. Follow with a third line element constructed at x2, y2 and continue it a short distance. The process proceeds iteratively and concludes when enough of the solution curve has been drawn to meet the needs of those concerned with the problem.

Numerical Methods Stability The constant h in Equations In general, the smaller the step-size, the more accurate the approximate solution becomes at the price of more work to obtain that solution. If h is chosen too large, then the approximate solution may not resemble the real solution at all, a condition known as numerical instability. General Remarks Regarding Numerical Methods A numerical method for solving an initial-value problem is a procedure that produces approximate solutions at particular points using only the operations of addition, subtraction, multiplication, division, and functional evaluations.

Each numerical method will produce approximate solutions at the points x0,x1,x2, Remarks made previously in this chapter on the step-size remain valid for all the numerical methods presented. The approximate solution at xn will be designated by y xn , or simply yn. The true solution at xn will be denoted by either Y xn or Yn.

Note that once yn is known, Equation In general, the corrector depends on the predicted value. The resulting equations are: It then follows from Equation Numerical Methods This is not a predictor-corrector method. The other three starting values are gotten by the Runge-Kutta method.

Order of a Numerical Method A numerical method is of order n, where n is a positive integer, if the method is exact for polynomials of degree n or less. In other words, if the true solution of an initial-value problem is a polynomial of degree n or less, then the approximate solution and the true solution will be identical for a method of order n.

In general, the higher the order, the more accurate the method. Generalizations to systems of three equations in standard form As in the previous section, four sets of starting values are required for the Adams-Bashforth-Moulton method.

Two solution curves are also shown, one that passes through the point 0,0 and a second that passes through the point 0,2. Numerical Methods Solved Problem Furthermore, it is assumed that a1 and b1 are not both zero, and also that a2 and b2 are not both zero.

The boundary-value problem is said to be homogeneous if both the differential equation and the boundary conditions are homogeneous i. Otherwise the problem is nonhomogeneous. In other words, a nonhomogeneous problem has a unique solution when and only when the associated homogeneous problem has a unique solution. Eigenvalue Problems When applied to the boundary-value problem You Need to Know Those values of l for which nontrivial solutions do exist are called eigenvalues; the corresponding nontrivial solutions are called eigenfunctions.

Form This condition can always be forced by multiplying Equation Properties of Sturm-Liouville Problems Property The eigenvalues of a Sturm-Liouville problem are real and nonnegative. Property The function w x in The basic features of all such expansions are exhibited by the trigonometric series discussed below.

Substituting these functions into This is a nonhomogeneous boundary-value problem of forms