Nonhomogeneous equations

Previously, some brief theory and a few examples were given that demonstrate the approach necessary to construct solutions to the homogeneous problem. The summarized procedure for the constant coefficient case is very simple, and one would like to maintain that simplicity for the case of at least the nonhomogeneous constant coefficient problem. 

The following two theorems are needed to establish the procedure for solving Equation 3.36  

Given one solution jp of the nonhomogeneous linear differential equation 

The above equation is the general solution of the nonhomogeneous equation 3.36. That is, the general solution y, of the nonhomogeneous equation is 

A relatively simple procedure for finding the most suitable candidate for yp is demonstrated below for the constant coefficient case. 

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The problem of n linear independent nonhomogeneous equations in n real unknowns... 

To select one from among the infinite number of solution sets, we must have an additional independent nonhomogeneous equation. If the additional equation is... 

This implies the estimate for a solution of the nonhomogeneous equation BVt +Ay = ip ... 

This system of n linear nonhomogeneous equations in n — 1 unknown quan-... 

Substituting C"(x) and C p x) into the nonhomogeneous equation, one obtains a system of first-order differential equations for a(x) and P(x) ... 

Nonhomogeneous equations with homogeneous boundary conditions. The... 

DIMENSIONAL EQUATIONS. Equations derived by empirical methods, in which experimental results are correlated by empirical equations without regard to dimensional consistency, usually are not dimensionally homogeneous and contain terms in several different units. Equations of this type are dimensional equations, or dimensionally nonhomogeneous equations. In these equations there is no advantage in using consistent units, and two or more length units, e.g., inches and feet, or two or more time units, e.g., seconds and minutes, may appear in the same equation. For example, a formula for the rate of heat loss from a horizontal pipe to the atmosphere by conduction and convection is... 

Equation 11.16 forms a system of M linear nonhomogeneous equations for the M unknowns a L=, m and can be solved by using any of the standard methods (e.g., Kramer s rule). In matrix notation, the solution is... 

Nonconstant density, 136 Nonhomogeneous equation, 117 Nonideal mixtures, 82 Nonisothermal pellet, 393 Nonlinear differential equation, 119 nth-order... 

Another example consists of the linear second-order nonhomogeneous equation... 

The solution of the corresponding homogeneous equation can be used to solve the nonhomogeneous equation. This solution is just the derivative d y of the solution of the nonlinear equation from the first step. Hence, the solution reads... 

To illustrate the solution of nonhomogeneous equations using Eq. (A.51), let us suppose we have a system equation (A.50) with A the same as the previous example. 

Moreover, the sediment balance that leads to the one-line governing equation, only accounts for longitudinal transport induced by wave breaking. To include the effect of other sources of sediment (e.g., artificial nom-ishment or river discharge) a nonhomogeneous equation has to be solved. 

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