This seems to be a second order total diferential equation - having T (temperature?) as a function and r as a parameter. You can look for the general solution in a diferential equation handbook, but you won't like the expression of the solution. If you want to get closer to the result, you would have to find the expression of the T function. Further improuvement could be obtained from expressing the boundary conditions - and so you get your result - the variance of T with the parameter r. If you only want to find the derivative of T, you also need it's expression. Regards!
I think that's a 2nd order Euler equation, whose solution can be found in a rather straightforward fashion. The first step is to multiply all terms by r^2. In this way you'll go from:
-(d2T/dr2)+(1/r)*(dT/dr)=0
to:
-r2*(d2T/dr2)+(r)*(dT/dr)=0
The second step is posing r=exp(x), and thus dr/dx=exp(x), x=ln(r), dx/dr=1/r=exp(-x). Just apply the substitution of the variables and resolve some passages (I'll skip them here). If a did those correctly, you should get to the form:
-exp(2x)*[exp(-x)*(d2T/dx2)*(dx/dr)+(dt/dx)*(-exp(-x))*(dx/dr)]+exp(x)*(dT/dx)*(dx/dr)=0
Since dx/dr=exp(-x), the final form of the equation becomes:
d2T/dx2+2dT/dx=0, whose characteristic equation is t2+2t=0 and has t=0 and t=-2 as solutions.
Hence, the general solution of the equation is:
T=C*exp(0*x)+D*exp(-2*x)=C+D/r2
Of course, you'll have to find C and D based on your boundary conditions.
Hope this helps. Best Regards,
Emiliano
This was of valuable help Mr.Emiliano and I could truly derive the functional using the above method. Thank u very much sir
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