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Chapter 2: Problem 129

What is the difference between an algebraic equation and a differential equation?

Short Answer

Expert verified
#tag_title# Quick Summary #tag_content# In conclusion, algebraic equations are mathematical equations involving variables and arithmetic operations without any derivatives. Differential equations, on the other hand, involve an unknown function and one or more of its derivatives. The key difference between the two lies in the presence of derivatives, and the solutions of differential equations can be both functions and numerical values, while algebraic equations usually have a finite number of numerical solutions.

Step by step solution

01

Definition of Algebraic Equation

An algebraic equation is a mathematical equation involving one or more variables and various arithmetic operations such as addition, subtraction, multiplication, and division, as well as powers and roots. No derivatives are involved in algebraic equations. For example, a quadratic equation is an algebraic equation: ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable.
02

Definition of Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. In other words, it is an equation that involves an unknown function and one or more of its derivatives with respect to one or more variables. Differential equations have applications in various fields like physics, engineering, and economics. An example of a simple first-order differential equation is: dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x, and y is the unknown function.
03

Key Differences

The main difference between algebraic and differential equations lies in the presence of derivatives. Algebraic equations do not have any derivatives and involve only arithmetic operations, whereas differential equations involve one or more derivatives of the unknown function. Another difference is that algebraic equations have a finite number of solutions, while differential equations may have infinite solutions, and sometimes these solutions are functions instead of specific numerical values.
04

Examples of Algebraic and Differential Equations

Here are a couple of examples to illustrate the difference between algebraic and differential equations. Algebraic Example: 2x^3 - 5x^2 + 3x + 6 = 0 This is an algebraic equation, and in this case, it's a cubic equation involving the variable x. Differential Example: d^2y/dx^2 - 3(dy/dx) + 2y = 0 This is a second-order linear homogeneous differential equation where the unknown function is y(x), and it involves its first and second derivatives.

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Most popular questions from this chapter

A spherical container of inner radius \(r_{1}=2 \mathrm{~m}\), outer radius \(r_{2}=2.1 \mathrm{~m}\), and thermal conductivity \(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is filled with iced water at \(0^{\circ} \mathrm{C}\). The container is gaining heat by convection from the surrounding air at \(T_{\infty}=25^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=18 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming the inner surface temperature of the container to be \(0^{\circ} \mathrm{C},(a)\) express the differential equation and the boundary conditions for steady one- dimensional heat conduction through the container, \((b)\) obtain a relation for the variation of temperature in the container by solving the differential equation, and \((c)\) evaluate the rate of heat gain to the iced water.

Consider a function \(f(x)\) and its derivative \(d f l d x\). Does this derivative have to be a function of \(x\) ?

A spherical shell, with thermal conductivity \(k\), has inner and outer radii of \(r_{1}\) and \(r_{2}\), respectively. The inner surface of the shell is subjected to a uniform heat flux of \(\dot{q}_{1}\), while the outer surface of the shell is exposed to convection heat transfer with a coefficient \(h\) and an ambient temperature \(T_{c \infty}\). Determine the variation of temperature in the shell wall and show that the outer surface temperature of the shell can be expressed as \(T\left(r_{2}\right)=\left(\dot{q}_{1} / h\right)\left(r_{1} / r_{2}\right)^{2}+T_{\infty \text { co }}\).

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

A long homogeneous resistance wire of radius \(r_{o}=\) \(5 \mathrm{~mm}\) is being used to heat the air in a room by the passage of electric current. Heat is generated in the wire uniformly at a rate of \(5 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\) as a result of resistance heating. If the temperature of the outer surface of the wire remains at \(180^{\circ} \mathrm{C}\), determine the temperature at \(r=3.5 \mathrm{~mm}\) after steady operation conditions are reached. Take the thermal conductivity of the wire to be \(k=6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\).
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