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Chapter 2: Problem 123

Can a differential equation involve more than one independent variable? Can it involve more than one dependent variable? Give examples.

Short Answer

Expert verified
Answer: Yes, a differential equation can involve more than one dependent variable, as shown in the example of a system of 2 ordinary differential equations: \( \frac{dy_1}{dt} = y_1^2 + y_2^2 \) and \( \frac{dy_2}{dt} = y_1 + y_2 \), where \( y_1 = y_1(t) \) and \( y_2 = y_2(t) \) are both dependent variables and t is the independent variable. Moreover, a differential equation can also involve more than one independent variable. This type of equation is known as a partial differential equation (PDE). An example is the 2D heat equation, which is a PDE with two independent variables x and y: \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \), where u = u(x, y) is the dependent variable and x, y are independent variables.

Step by step solution

01

Dependent variables in differential equations

A differential equation can indeed involve more than one dependent variable. For example, consider a system of 2 ordinary differential equations: \( \frac{dy_1}{dt} = y_1^2 + y_2^2 \) and \( \frac{dy_2}{dt} = y_1 + y_2 \), where \( y_1 = y_1(t) \) and \( y_2 = y_2(t) \) are both dependent variables and t is the independent variable.
02

Independent variables in differential equations

A partial differential equation (PDE) is an equation involving multiple independent variables and their partial derivatives. For example, a 2D heat equation is a PDE having two independent variables x and y: \( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \), where u = u(x, y) is the dependent variable and x, y are independent variables.

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Most popular questions from this chapter

A solar heat flux \(\dot{q}_{s}\) is incident on a sidewalk whose thermal conductivity is \(k\), solar absorptivity is \(\alpha_{s}\), and convective heat transfer coefficient is \(h\). Taking the positive \(x\) direction to be towards the sky and disregarding radiation exchange with the surroundings surfaces, the correct boundary condition for this sidewalk surface is (a) \(-k \frac{d T}{d x}=\alpha_{s} \dot{q}_{s}\) (b) \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)\) (c) \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)-\alpha_{s} \dot{q}_{s}\) (d) \(h\left(T-T_{\infty}\right)=\alpha_{s} \dot{q}_{s}\) (e) None of them

A spherical metal ball of radius \(r_{o}\) is heated in an oven to a temperature of \(T_{i}\) throughout and is then taken out of the oven and allowed to cool in ambient air at \(T_{\infty}\) by convection and radiation. The emissivity of the outer surface of the cylinder is \(\varepsilon\), and the temperature of the surrounding surfaces is \(T_{\text {surr }}\). The average convection heat transfer coefficient is estimated to be \(h\). Assuming variable thermal conductivity and transient one-dimensional heat transfer, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

What is the difference between an ordinary differential equation and a partial differential equation?

Consider a spherical shell of inner radius \(r_{1}\) and outer radius \(r_{2}\) whose thermal conductivity varies linearly in a specified temperature range as \(k(T)=k_{0}(1+\beta T)\) where \(k_{0}\) and \(\beta\) are two specified constants. The inner surface of the shell is maintained at a constant temperature of \(T_{1}\) while the outer surface is maintained at \(T_{2}\). Assuming steady one- dimensional heat transfer, obtain a relation for \((a)\) the heat transfer rate through the shell and ( \(b\) ) the temperature distribution \(T(r)\) in the shell.

Consider a water pipe of length \(L=17 \mathrm{~m}\), inner radius \(r_{1}=15 \mathrm{~cm}\), outer radius \(r_{2}=20 \mathrm{~cm}\), and thermal conductivity \(k=14 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Heat is generated in the pipe material uniformly by a \(25-\mathrm{kW}\) electric resistance heater. The inner and outer surfaces of the pipe are at \(T_{1}=60^{\circ} \mathrm{C}\) and \(T_{2}=80^{\circ} \mathrm{C}\), respectively. Obtain a general relation for temperature distribution inside the pipe under steady conditions and determine the temperature at the center plane of the pipe.
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