Equations with Separable Variables

In this section, we discuss equations that can be manipulated algebraically into the form 

If we have manipulated the equation into the form of Eq. (8.62), we say that we have separated the variables, because we have no x dependence in the left-hand side of the equation and no y dependence in the right-hand side. We can perform 

EXAMPLE 8.6 In afirst-order chemical reaction with no back reaction, the concentration of the reactant is governed by 

In the last step, we recognized that had to equal the concentration at time t = 0. A definite integration can be carried out instead of an indefinite integration  

This equation is the same as Eq. (8.67) except that the time is now called t instead off. The limits on the two definite integrations must be done correctly. If the lower limit of the time integration is zero, the lower limit of the concentration integration must be the value of the concentration at zero time. The upper limit is similar.  

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Equations with Separable Variables Every differential equation of the first order and of the first degree can be written in the form M x, y) dx + N x, y) dy = 0. If the equation can be transformed so that M does not involve y and N does not involve x, then the variables are said to be separated. The solution can then be obtained by quadrature, which means that y = f f x) dx + c, which may or may not be expressible in simpler form. 

Sometimes an equation out of this classification can be altered to fit by change of variable. The equations with separable variables are solved with a table of integrals or by numerical means. Higher order linear equations with constant coefficients are solvable with the aid of Laplace Transforms. Some complex equations may be solvable by series expansions or in terms of higher functions, for instance the Bessel equation encountered in problem P7.02.07, or the equations of problem P2.02.17. In most cases a numerical solution Is possible. 

One can readily integrate equation (3.2.7), since the substitution W = dc/dz results in a first-order equation with separating variables. 

The Lotka equations can be solved exactly. Dividing the two equations we obtain an equation with separated variables which can be integrated, yielding the constant of motion H(x, y)... 

Section 8.3 Differential Equations with Separable Variables... 

Relation a(f) is obtained by the integration of the preceding differential equation with separate variables ... 

This type of a pattern of singular points is called a centre - Fig. 2.3. A centre arises in a conservative system indeed, eliminating time from (2.1.28), (2.1.29), one arrives at an equation on the phase plane with separable variables which can be easily integrated. The relevant phase trajectories are closed the model describes the undamped concentration oscillations. Every trajectory has its own period T > 2-k/ujq defined by the initial conditions. It means that the Lotka-Volterra model is able to describe the continuous frequency spectrum oj < u>o, corresponding to the infinite number of periodical trajectories. Unlike the Lotka model (2.1.21), this model is not rough since... 

Remember 2.4 The method for numerical solution of differential equations with complex variables is to separate the equations into real and imaginary parts and to solve them simultaneously. 

Instead of solving one equation with N variables, the problem is transformed into solving N equations with only one variable. When the differential operator is transformed into a matrix representation, the separation is equivalent to finding a coordinate system where the representation is diagonal. 

Many of the fundamental equations in physics (and science in general) are formulated as differential equations. Typically, the desired mathematical function is known to obey some relationship in terms of its first and/or second derivatives. The task is to solve this differential equation to find the function itself. A complete treatment of the solution of differential equations is beyond the scope of this book, and only a simplified introduction is given here. Furthermore, we will only discuss solutions of differential equations with one variable. In most cases the physical problem gives rise to a differential equation involving many variables, but prior to solution these can often be (approximately) decoupled by separation of the variables, as discussed in Section 1.6. 

Till now we have used only symbolic resources of Mathcad system to solve differential equations. We again accentuate, that all kinetic equations from above are the equations with separatable variables. User need to separate those variables without assistance to obtain a solution. And getting final solution is in fact an integration of both parts of the obtained equalities. 

The variables are separable, but an integration in closed form is not possible because of the odd exponent. Numerical integration followed by substitution into (4) will provide both A and B as functions of t. The plots, however, are of solutions of the original differential equations with ODE. 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

Equation 5.2.59 simplifies greatly when m1 = m2 (i.e., both reactions are the same order with respect to A) because it is then possible to separate variables and integrate each term directly. Reactions of this type are the only ones that we will consider in more detail. 

The wave equation is a second-order partial differential equation in three variables. The usual technique for solving such an equation is to use a procedure known as the separation of variables. However, with r expressed as the square root of the sum of the squares of the three variables, it is impossible... 

The SP-DFT has been shown to be useful in the better understanding of chemical reactivity, however there is still work to be done. The usefulness of the reactivity indexes in the p-, p representation has not been received much attention but it is worth to explore them in more detail. Along this line, the new experiments where it is able to separate spin-up and spin-down electrons may be an open field in the applications of the theory with this variable set. Another issue to develop in this context is to define response functions of the system associated to first and second derivatives of the energy functional defined by Equation 10.1. But the challenge in this case would be to find the physical meaning of such quantities rather than build the mathematical framework because this is due to the linear dependence on the four-current and external potential. 

When combined with the Fourier expansion of functions, separation of variables is another powerful method of solutions which is particularly useful for systems of finite dimensions. Regardless of boundary conditions, we decompose the solution C(x, t), where the dependence of C on x and t is temporarily emphasized, to the general one-dimensional diffusion equation with constant diffusion coefficient... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

There is an important disadvantage stemming from variable separation and use of the FFT, however. One is limited to the use of the point simultaneous procedure in the determination of the coefficients. That is, all of the coefficients must be computed in an iteration before they can be resubstituted back in the iterative equations. With the point successive method, the improved coefficient determined by one equation is substituted in the succeeding equation to render additional improvement. With the test data treated in this work, the point simultaneous procedure converged much more slowly than the point successive method. 

Even though the problem asks for Libin s age, it ll be easier to let the variable, x, represent Larry s age and then answer the question after solving the equation. Letting x represent Larry s age, then 3x represents Libin s age. The sum of their ages is x + 3x. You write 16 more than twice Larry s age plus 4 with numbers and a variable in exactly that same order 16 + 2x + 4. Now write the entire equation with the two expressions separated by an equal sign and solve. 

Many word problems lend themselves to more than one equation with more than one variable. It s easier to write two separate equations, but it takes more work to solve them for the unknowns. And, in order for there to be a solution at all, you have to have at least as many equations as variables. 

Physics texts often introduce spherical harmonics by applying the technique of separation of variables to a differential equation with spherical symmetry. This technique, which we will apply to Laplace s equation, is a method physicists use to hnd solutions to many differential equations. The technique is often successful, so physicists tend to keep it in the top drawer of their toolbox. In fact, for many equations, separation of variables is guaranteed to find all nice solutions, as we prove in Proposition A.3. 

The right-hand side is constant in r, so the left-hand side must also be constant in r. Contrariwise, both sides are constant in 3 and (p. In other words, the variables are separated into different terms, a happy accident that we can exploit. We started with one differential equation involving three variables, and ended up with two separate equations, one involving one variable, and one involving two variables. Thus we have reduced the problem (finding solutions to the original equation) to two simpler problems. Of course, this simplification works only if our supposition (that there are solutions of the given form) turns out to be true. 

The curve fitting programs cope better with fewer variables in the equations. Try to reduce the number of variables. For example, suppose you have to fit a multiphasic curve to three exponentials that are moderately separated in time. There are seven unknowns three rate constants three amplitudes and an endpoint. If the slowest phase is sufficiently separated from the second, first fit the tail of the slowest phase to a single exponential. Then fit the whole curve to a triple exponential equation in which the rate constant and the amplitude that were derived for the third phase are used as constants. Use a time window that focuses on the first two phases and not the whole time course. Similarly, if the first phase is much faster than the second and third, fit the tail of the process to two exponentials. Then fit the fast time region to a triple exponential in which the last two phases have fixed rate constants and amplitudes. 

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