Respuesta :
Answer:T=[tex]\frac{-26}{r}[/tex]+36
Explanation:
Given Temperature at r=1m is [tex]10^{\circ}C [/tex]
Temperature at r=2m is [tex]28^{\circ}C[/tex]
[tex]\frac{\mathrm{d^2} T}{\mathrm{d} r^2}+\frac{2}{r}\frac{\mathrm{d} T}{\mathrm{d} r}=0[/tex]
Let [tex]\frac{2}{r}\frac{\mathrm{d} T}{\mathrm{d} r}=S[/tex]
[tex]\frac{\mathrm{d^2} T}{\mathrm{d} r^2}=\frac{2}{r}\frac{\mathrm{d} S}{\mathrm{d} r}[/tex]
therefore [tex]\frac{2}{r}\frac{\mathrm{d} S}{\mathrm{d} r}+\frac{2}{r}[/tex]S=0
[tex]\frac{2}{r}\frac{\mathrm{d} S}{\mathrm{d} r}=-\frac{-2S}{r}[/tex]
solving
[tex]r^2[/tex] S=constant
substitute S value
[tex]\frac{\mathrm{d}T}{\mathrm{d}r}[/tex]=[tex]\frac{c}{r^2}[/tex]
Solving it we get
T=[tex]\frac{-c}{r}+c_2[/tex]
Now using given condition
10=[tex]\frac{-c}{1}+c_2[/tex]
28=[tex]\frac{-c}{2}+c_2[/tex]
[tex]c_2[/tex]=36,c=26
putting c values
T=[tex]\frac{-26}{r}[/tex]+36
The expression for the temperature T(r) for the two spheres one inside other is given as,
[tex]T=-\dfrac{26}{r}+36[/tex]
What is differential equation?
Differential equation is the equation in which there is one or more number of unknown variable exist to find the rate of change of one variable with respect to other.
The radius of the sphere is 1 m and the temperature of the sphere is 10 degree Celsius. The sphere lies inside a concentric sphere with radius 2 m and temperature 28°C.
The differential equation which spheres satisfies is given as,
[tex]\dfrac{d^2T}{dr^2}+\dfrac{2}{r}\dfrac{dT}{dr}=0[/tex]
Let the above equation is equation one,
Suppose S = dT/dr, then
[tex]\dfrac{dS}{dt}=\dfrac{2}{r}\dfrac{d^2T}{dr^2}[/tex]
Put the values in the equation one as,
[tex]\dfrac{2}{r}\dfrac{dS}{dt}+\dfrac{-2}{r}S=0\\\dfrac{2}{r}\dfrac{dS}{dt}=-\dfrac{-2}{r}S\\r^2S=C[/tex]
Here, (C) is the constant value. Now put the value of S as,
[tex]r^2\dfrac{dT}{dr}=C\\\dfrac{dT}{dr}=\dfrac{C}{r^2}\\T=-\dfrac{C}{r}+C_2[/tex]
Put the value of radius as 1 m,
[tex]10=-\dfrac{C}{1}+C_2\\10=-C+C_2[/tex]
Similarly, for the radius 2 meters and temperature 28 degree Celsius,
[tex]28=-\dfrac{C}{2}+C_2[/tex]
On solving above equation, we get,
[tex]C=26\\C_2=36[/tex]
Put the values of constant for the required expression as,
[tex]T=-\dfrac{26}{r}+36[/tex]
Thus, the expression for the temperature T(r) for the two spheres one inside other is given as,
[tex]T=-\dfrac{26}{r}+36[/tex]
Learn more about the differential equation here;
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