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Chapter 2: Problem 133

How do you recognize a linear homogeneous differential equation? Give an example and explain why it is linear and homogeneous.

Short Answer

Expert verified
A linear homogeneous differential equation is an equation with no products of the dependent variable and its derivatives and has a 0 on the right-hand side. For example, the equation \(\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0\) is a linear homogeneous differential equation because it is linear and has a 0 on the right-hand side.

Step by step solution

01

Definition of a Linear Homogeneous Differential Equation

A Linear Homogeneous Differential Equation is an equation of the form: \[a_n(x)\frac{d^ny}{dx^n}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_1(x)\frac{dy}{dx}+a_0(x)y=g(x)\] where \(a_n(x), a_{n-1}(x), \dots , a_1(x), a_0(x)\) are continuous functions on a given interval, and \(g(x) = 0\), indicating that it is homogeneous. Now, let's provide an example and explain why it is linear and homogeneous.
02

Providing an Example

Let's consider the following example of a differential equation: \[\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0\]
03

Checking Linearity

To check if the given example is linear, we will verify if the differential equation has no product of the dependent variable \(y\) and its derivatives. In our example: \[\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0\] we can see that there are no products of \(y\) and its derivatives. Thus, the differential equation is linear.
04

Checking Homogeneity

To check if the given differential equation is homogeneous, we need to verify that it has a \(0\) on the right-hand side. In our example: \[\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0\] we can see that the right-hand side is indeed \(0\), so the differential equation is homogeneous.
05

Conclusion

In conclusion, the given example \(\frac{d^2y}{dx^2}+3\frac{dy}{dx}+2y=0\) is a linear homogeneous differential equation because it is a linear equation with no products of the dependent variable and its derivatives, and it is homogeneous since the right-hand side is \(0\).

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Most popular questions from this chapter

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 dropped into a large body of water at \(T_{\infty}\) where it is cooled by convection with an average convection heat transfer coefficient of \(h\). Assuming constant 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.

Consider a steam pipe of length \(L=35 \mathrm{ft}\), inner radius \(r_{1}=2\) in, outer radius \(r_{2}=2.4\) in, and thermal conductivity \(k=8 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). Steam is flowing through the pipe at an average temperature of \(250^{\circ} \mathrm{F}\), and the average convection heat transfer coefficient on the inner surface is given to be \(h=\) \(15 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\). If the average temperature on the outer surfaces of the pipe is \(T_{2}=160^{\circ} \mathrm{F},(a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the pipe, \((b)\) obtain a relation for the variation of temperature in the pipe by solving the differential equation, and \((c)\) evaluate the rate of heat loss from the steam through the pipe.

A large plane wall, with a thickness \(L\) and a thermal conductivity \(k\), has its left surface \((x=0)\) exposed to a uniform heat flux \(\dot{q}_{0}\). On the right surface \((x=L)\), convection and radiation heat transfer occur in a surrounding temperature of \(T_{\infty}\). The emissivity and the convection heat transfer coefficient on the right surface are \(\bar{\varepsilon}\) and \(h\), respectively. Express the houndary conditions and the differential equation of this heat conduction problem during steady operation.

Consider a long solid cylinder of radius \(r_{o}=4 \mathrm{~cm}\) and thermal conductivity \(k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Heat is generated in the cylinder uniformly at a rate of \(\dot{e}_{\text {gen }}=35 \mathrm{~W} / \mathrm{cm}^{3}\). The side surface of the cylinder is maintained at a constant temperature of \(T_{s}=80^{\circ} \mathrm{C}\). The variation of temperature in the cylinder is given by $$ T(r)=\frac{\dot{e}_{\text {gen }} r_{o}^{2}}{k}\left[1-\left[1-\left(\frac{r}{r_{o}}\right)^{2}\right]+T_{s}\right. $$ Based on this relation, determine \((a)\) if the heat conduction is steady or transient, \((b)\) if it is one-, two-, or three-dimensional, and \((c)\) the value of heat flux on the side surface of the cylinder at \(r=r_{o^{*}}\)

A large steel plate having a thickness of \(L=4\) in, thermal conductivity of \(k=7.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\), and an emissivity of \(\varepsilon=0.7\) is lying on the ground. The exposed surface of the plate at \(x=L\) is known to exchange heat by convection with the ambient air at \(T_{\infty}=90^{\circ} \mathrm{F}\) with an average heat transfer coefficient of \(h=12 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) as well as by radiation with the open sky with an equivalent sky temperature of \(T_{\text {sky }}=480 \mathrm{R}\). Also, the temperature of the upper surface of the plate is measured to be \(80^{\circ} \mathrm{F}\). Assuming steady onedimensional heat transfer, \((a)\) express the differential equation and the boundary conditions for heat conduction through the plate, \((b)\) obtain a relation for the variation of temperature in the plate by solving the differential equation, and \((c)\) determine the value of the lower surface temperature of the plate at \(x=0\).
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