Ordinary differential equations separable variables

Separation of Variables This is a powerful, well-utilized method which is applicable in certain circumstances. It consists of assuming that the solution for a partial differential equation has the form U =f x)g(y)- If it is then possible to obtain an ordinary differential equation on one side of the equation depending only on x and on the other side only on y, the partial differential equation is said to be separable in the variables x, y. If this is the case, one side of the equation is a function of x alone and the other of y alone. The two can be equal only if each is a constant, say X. Thus the problem has again been reduced to the solution of ordinaiy differential equations. 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

A partial differential equation is one with two or more independent variables. The separation of these variables, if it can be carried out, yields ordinary differential equations which can, in most cases, be solved by the various methods presented in Chapters 3 and 5. The general approach to this problem will now be illustrated by a number of examples that are fundamental in physics and chemistry. 

As before, we examine the time evolution of the shape-preserving solution by separating variables. The ordinary differential equation for the time evolution, dw/dt = -aw yields w(t) = (1 + t/x), which is the functional form we use to fit the relaxation... 

A solution to equation (E2.1.2) may be achieved by (1) separating variables and integrating or (2) solving the equation as a second-order, linear ordinary differential equation. We will use the latter because the solution technique is more general. 

Under conditions of constant shear, dy /dt = 0, Eq. (5.67) becomes an ordinary differential equation, which can be solved by separation of variables and integration using the boundary conditions r = tq at t = 0 and r = r at t = t to give the following relation for the shear stress, r, as a function of time... 

Reversible bimolecular reactions such asA + B C + D can be solved exactly by the method of separation of variables and the ordinary differential equations in the variable s are Lame equations. This makes the evaluation of the Fourier-type coefficients very difficult since derivative formulas and orthogonality conditions do not seem to exist or at least are not easily used. In addition to this, even if such formulas did exist, it seems unlikely that numerical results could be easily obtained. It does turn out, however, that these reversible bimolecular processes can be solved exactly and conveniently in the equilibrium limit, and this was done by Darvey, Ninham, and Staff.14... 

There are several ways to solve a third-order ordinary-differential-equation boundary-value problem. One is shooting, which is discussed in Section 6.3.4.1. Here, we choose to separate the equation into a system of two equations—one second-oider and one first-order equation. The two-equation system is formed in the usual way by defining a new variable g = /, which itself serves as one of the equations,... 

Cylindrical Coordinates. The separation-of-variables method also applies when the boundary conditions and initial conditions have cylindrical symmetry (see Eqs. 5.7 and 5.8). If c(r, t) = R r)T t), the resulting ordinary differential equation for R(r) is... 

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

A different classification scheme for DEs, short for differential equations, separates those DEs with a single independent variable dependence, such as only time or only 1-dimensional position, from those depending on several variables, such as time and spatial position. DEs involving a single independent variable are routinely called ODEs, or ordinary differential equations. DEs involving several independent variables such as space and time are called PDEs, or partial differential equations because they involve partial derivatives. 

To solve the 2nd order ordinary differential equation in Rqs. 4-39, wc define a new variable w as w = dT/dj). This reduces Eq. 4-39u into a first order differential equation than can be solved by separating variables. 

Each of the first three terms in Eq (3.40) depends on one variable only, independent of the other two. This is possible only if each term separately equals a constant, say, —a, and — respectively. These constants must be negative in order that > 0. Eq (3.40) is thereby transformed into three ordinary differential equations ... 

Partial differential equations are generally solved by finding a transformation tiiat allows the partial differential equation to be converted into two ordinary differential equations. A number of techniques are available, including separation of variables, Laplace transforms, and the method of characteristics. 

The left hand side of (2.164) depends on the (dimensionless) time f+, the right hand side on the position coordinate r+ the variables are separated. The equality demanded by (2.164) is only possible if both sides of (2.164) are equal to a constant —fi2. This constant /( is known as the separation parameter. With this the following ordinary differential equations are produced from (2.164)... 

The alert reader will notice that although the left-hand-side of this equation depends only on x, the right-hand-side depends only on t. So both sides must be equal to the same constant. Now you have two easy ordinary differential equations in one unknown each. Also you have an unidentified flying parameter, namely the constant that both sides of the equation must equal. In the grand tradition of calculus textbooks, let us call this constant C. So now we have two separate equations to deal with, each in only one variable. The first one is ... 

The separation of variables is a common technique used to solve linear PDEs. This technique will be discussed in detail in chapter 7. This technique yields ordinary differential equations for the eigenfunctions. In this section, we will present two numerical techniques for the Graetz problem. 

As a result of our separation of variables strategy, the problem of solving the Schrodinger equation for the quantum corral is reduced to that of solving two ordinary differential equations. In particular, we have... 

The solution to this ordinary differential equation may be written down by separation of variables as... 

In fact, the ordinary differential equations in each variable from such a separation have the standard form of the Lam6 equation [12] ... 

They admit factorizable solutions Eq. (43) and can be separated into ordinary differential equations in each variable, provided the geometric and dynamic parameters are related through Eq. (44), so that... 

This equation differs from Equation 3 in that the flow velocity u is given by u = 2u0 (1 — r2/r02) u0 is as before the average flow velocity, and r0 is the tube radius. As usual we let n(r,x) = X(x)R(r), substitute into Equation A-l, separate variables, and obtain two ordinary differential equations for X(x) and R(r), which are related by a separation constant b2. 

Substituting the expansion (3.4.12) into (3.4.1) and then separating the variables, we obtain the following ordinary differential equation for the functions Hm ... 

As we learned in this chapter, the formulation of unsteady distributed problems leads to partial differential equations. The solution of these equations is much more involved than that of ordinary differential equations. Among the techniques available, the analytical and computational methods are most frequently referred to. Exact analytical methods such as separation of variables and transform calculus are beyond the scope of the text. However, the method of complex temperature and the use of charts based on exact analytical solutions, being useful for some practical problems, are respectively discussed in Sections 3.4 and 3.6. Among approximate analytical methods, the integral method, already introduced in Sections 2.4 and 3.1, is further discussed in Section 3.5. The analog solution technique is also briefly treated in Section 3.7. 

The assumption that the wavefunction can be written as a product of single-variable functions is a valid one, for we can find ordinary differential equations for the assumed factors. That is what it means for a partial differential equation to be separable. 

The separation of variables is complete, and we have two ordinary differential equations. Except for the symbols used, both of these equations are the same as... 

Many partial differential equations arising in physical problems can be solved by separation of variables. In this procedure, a trial solution consisting of factors depending on one variable each is introduced, and the resulting equation is manipulated until the variables occur only in separate terms. Setting these terms equal to constants gives one ordinary differential equation for each variable. 

For other types of patterns, we recall that the complete governing equation (or set of equations) is (are) reduced to an ordinary differential equation involving z only, by means of separation of variables. However, the equation in X and y, which was set aside since we considered only the equation in z for determining the critical stability parameter, also involved a and became... 

Substitution of (13.37) into eiordinary differential equations, one for L(0 and one for M rj). Solving these equations, one finds that the condition that be well-behaved requires that, for each fixed value of R, only certain values of E i are allowed this gives a set of different electronic states. There is no algebraic formula for E h it must be calculated numerically for each desired value of R for each state. In addition to the quantum number m, the electronic wave functions are characterized by the quantum numbers and which give the number of nodes in the L( ) and M rf) factors in... 

There are three unknown independent variables in (3.2), namely x, y, z. One should separate the variables in order to split the partial differential equation into a set of three ordinary differential equations, each involving only one coordinate. However, the separation of variables cannot be carried out when rectangular coordinates are employed because the Coulomb potential energy (3.1) could not be represented as a product of functions each depending on one variable only. 

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