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A sphere with radius 1 m has temperature $ 15^oC. $ It lies inside a concentric sphere with radius 2 m and temperature $ 25^oC. $ The temperature $ T(r) $ at a distance $ r $ from the common center of the spheres satisfies the differential equation $ \frac {d^2T}{dr^2} + \frac {2}{r}\frac {dT}{dr} = 0 $If we let $ S = dT/dr, $ then $ S $ satisfies a first-order differential equation. Solve it to find an expression for the temperature $ T(r) $ between the spheres.

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$T(r)=\frac{-20}{r}+35$

Calculus 2 / BC

Chapter 9

Differential Equations

Section 3

Separable Equations

Oregon State University

University of Nottingham

Idaho State University

Lectures

13:37

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. An ordinary differential equation (ODE) is a differential equation containing one or more derivatives of a function and their rates of change with respect to the function itself; it can be used to model a wide variety of phenomena. Differential equations can be used to describe many phenomena in physics, including sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, quantum mechanics, and general relativity.

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A sphere with radius 1 $\m…

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A sphere with radius 1$\ma…

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Consider two concentric sp…

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