Why do Second Order Ordinary Differential Equations Occur?

Because the right hand sides of equations (1.1) , (1.6), and (1.11) are constants, differentiating each equation once yields a second order equation:
(2.1)

(2.2)

(2.3)

The boundary value problem (BVP) consisting of equation (2.1) and boundary conditions (1.2) is equivalent to the BVP consisting of equations (1.1) and (1.2). Solving (2.1) subject to the boundary conditions (1.2) gives the solution (1.3), exactly as before. Then applying Fourier's law, q = - k (dT/dx) , we obtain the expression (1.5) for the heat flux, as before. Similar remarks apply to the second order equations (2.2) and (2.3).

You should note that in each of Examples 1, 2b and 3, two boundary conditions were written (equations (1.2), (1.7) and (1.13)). This seems incorrect if we are solving first order equations. But the two boundary conditions allowed us not only to determine the temperature distribution but also a heat flux or flow rate (q or Q). If only one boundary condition is given, q or Q would have to be independently specified in the problem.

Since each of the these examples could be formulated in terms of a first order ordinary differential equation (ODE), you may wonder why we introduce ODE's of second order, equations (2.1), (2.2) and (2.3). Why introduce this complication? After all, as pointed out in the remarks following equation (2.3), in each of the examples 1, 2b and 3, the BVP's for the first order ODE and the associated second order ODE are equivalent when the same boundary conditions are used.

As an example, work Example Problem 2.1 following this page.

The point of introducing the second order DE's is that in general, heat conduction processes require DE's of second order for their mathematical description.

In Examples 1, 2b and 3, formulated above, heat enters the region of interest through one boundary and leaves through another. No heat is generated or lost in the interior of the region. This allows the use of a first order DE to describe each problem.

If however, there are sources or sinks within the region where heat transfer is under study (in other words, the region where the DE is to hold), the problem must be formulated in terms of a second order DE, as we illustrate in the next section.


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