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Chapter 2: Problem 136

How is the order of a differential equation determined?

Short Answer

Expert verified
Answer: The order of a differential equation is the highest power of the derivative present in the equation, or the highest number of times the dependent variable has been differentiated with respect to the independent variable. To determine the order of a differential equation, first identify the dependent variable (y or f(x)) and its derivatives present in the equation. Then, note the highest power (n) of the derivative present, which will give you the order of the differential equation. The order helps to classify the differential equation and provides insights into the nature of its solution(s).

Step by step solution

01

Understand the concept of a differential equation

A differential equation is an equation that includes one or more derivatives of a dependent variable (usually denoted as y or f(x)) with respect to an independent variable (usually denoted as x). The purpose of a differential equation is to find the function(s) that satisfy the given equation involving its derivatives.
02

Define the order of a differential equation

The order of a differential equation is the highest power of the derivative present in the equation. In other words, it is the highest number of times the dependent variable has been differentiated with respect to the independent variable in the given equation.
03

Determine the order of a differential equation

To determine the order of a differential equation, follow these steps: 1. Identify the dependent variable (y or f(x)) and its derivatives, if any, present in the equation. 2. Note the highest power (n) of the derivative present. This will give you the order of the differential equation. The order of a differential equation helps to classify the equation and provides insights into the nature of its solution(s).

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Most popular questions from this chapter

Consider a solid cylindrical rod whose side surface is maintained at a constant temperature while the end surfaces are perfectly insulated. The thermal conductivity of the rod material is constant, and there is no heat generation. It is claimed that the temperature in the radial direction within the rod will not vary during steady heat conduction. Do you agree with this claim? Why?

How do you distinguish a linear differential equation from a nonlinear one?

How do differential equations with constant coefficients differ from those with variable coefficients? Give an example for each type.

A heating cable is embedded in a concrete slab for snow melting. The heating cable is heated electrically with joule heating to provide the concrete slab with a uniform heat of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). The concrete has a thermal conductivity of \(1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). To minimize thermal stress in the concrete, the temperature difference between the heater surface \(\left(T_{1}\right)\) and the slab surface \(\left(T_{2}\right)\) should not exceed \(21^{\circ} \mathrm{C}\) (2015 ASHRAE Handbook-HVAC Applications, Chap. 51). Formulate the temperature profile in the concrete slab, and determine the thickness of the concrete slab \((L)\) so that \(T_{1}-\) \(T_{2} \leq 21^{\circ} \mathrm{C}\).

Consider a medium in which the heat conduction equation is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ (a) Is heat transfer steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the thermal conductivity of the medium constant or variable?
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