Homogeneous Linear Second-Order Differential Equations

HOMOGENEOUS LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS... 

It is a homogeneous, linear, second-order, differential equation defined in the complex plane. In the two-dimensional space of its particular solutions, one can choose a solution which is finite at jc = 0. Then the second linearly independent solution behaves atx = 0 as x ". The solution finite at a = 0 is usually denoted N(a, p, y, S-,x) and referred to as the biconfluent Heun function. It is usually expressed as [30]... 

Remember 2.2 The general solution to nonhomogeneous linear second-order differential equations with constant coefficients can be obtained as the product of function to be determined and the solution to the homogeneous equation (see equation (2.41)). 

In order to discuss the methods used to solve linear second-order differential equations, it is necessary to reintroduce the term homogeneous, or complementary, but with a meaning unrelated to previous usage in this book. 

Because our interest is with second-order differential equations, two linearly independent solutions always arise (the Wronskian of solutions is non-zero [490], see Sect. 5) and requires two arbitrary constants to be fixed from the two boundary conditions imposed on p0(r, t) by the physics of the problem being modelled. These boundary conditions determine how much of each of the two linearly independent solutions of the homogeneous equation (317) must be added to the particular integral to ensure that the solution of eqn. (316) is consistent with the boundary conditions. In the next three sections, the method of deriving the particular integral from the two linearly independent solutions of the homogeneous equation are discussed. 

Differential equations. A linear, homogeneous, second-order differential equation has the form... 

This can be differentiated once, to yield a linear homogeneous second-order differential equation ... 

Equation 3-56 is a linear, homogeneous, second-order differential equation with constant coefficients. A fundamental theory of differential equations states (hat such an equation has two linearly independent solution functions, and its general solution is the linear combination of those two solution functions. A careful examination of the differential equation reveals that subtracting a constant multiple of the solution function 0 from its second derivative yields zero. Thus we conclude that the function 0 and its second derivative must be constant multiples of each other. The only functions whose derivatives are constant multiples of the functions themselves are the exponential functions (or a linear combination of exponential functions such as sine and cosine hyperbolic functions). Therefore, the solution functions of the differential equation above are the exponential functions e or or constant multiples of them. This can be verified by direct substitution. For example, the second derivative of e is and its substitution into Eq. 3-56... 

By dividing by the coefficient of y", we can put any linear homogeneous second-order differential equation into the form... 

The application of the time-independent Schrodinger equation to a system of chemical interest requires the solution of a linear second-order homogeneous differential equation of the general form... 

Solution Equation (2.70) is a linear second-order homogeneous differential equation with constant coefficients. It can be solved using the characteristic equation... 

These differential equations are linear, homogeneous, and second order. The general solution of either is a linear combination of any two linearly independent solutions, and this fact is alone suffieient to show that the corresponding fields B z) and potentials 0(z) have an imaging action, as we now show. Consider the particular solution h z) of Eq. (10) that intersects the axis at z = Zo and z = Zi (Fig. 1). A pencil of rays that intersects the plane z = Zo at some point Po( o Jo) can be described by... 

The most general constant coefficient, linear, second-order, ordinary, homogeneous differential equation is... 

Consider the variable coefficient, linear second-order and homogeneous differential equation... 

To construct a solution to the homogeneous case, we begin with an approach similar to that used in the discussion of linear second-order constant coefficient differential equations. We seek solutions to Equation 3.156 of the form... 

The differential equation of (6.13.5) is a linear second-order, nonho-mogeneous equation. From Appendix A we find that the homogeneous solution is... 

Rearrange this linear second-order homogeneous ordinary differential equation to obtain... 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

This is a homogeneous linear differential equation of second order and its characteristic equation is... 

Equation (11.41) is a second-order, linear, and homogeneous differential equation. 

The general solution to the linear, homogeneous, second-order, constant-coefficients differential equation y" x) + py x) + qy x) = 0 is y = where Si and S2... 

So far we have considered only cases where the potential energy V(ac) is a constant. Hiis makes the SchrSdinger equation a second-order linear homogeneous differential equation with constant coefficients, which we know how to solve. However, we want to deal with cases in which V varies with x. A useful approach here is to try a power-series solution of the SchrSdinger equation. 

The linear homogenous differential equation of second order (E9) has a solution of the form. 

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