- #1
atta_bo-y
- 5
- 0
Hey guys,
I have a smallish problem with a differential equation.
Set up:
There is a sphere in a room; and it's being illuminated with intensity I from one side.
The room temperature (T_{environment}) is set to be constant.
[ latex ]
\Delta Q = Q_{in} - Q_{out}\\
= I\Delta t A - P_{out}\Delta t + P_{in}\Delta t\\
= I\Delta t A - \sigma A T^{4} \Delta t + \sigma A T^{4}_{environment} \Delta t\\
\Delta Q = cm\Delta T\\
\frac{\Delta T}{\Delta t} = I\Delta t \pi r^2 - \sigma 4 \pi r^2 T^{4} \Delta t + \sigma 4 \pi r^2 T^{4}_{environment} \Delta t
[ /latex ]
Hence we have a differential equation in the form of
!
[ latex ]
\frac{dT(t)}{dt} = a - b*T^4(t)
[ /latex ]
!
I have tried different methods... But none of them worked... (And wolframalpha can only solve for t :-( )
Thanks for your consideration ;-)
atta_bo-y
I have a smallish problem with a differential equation.
Set up:
There is a sphere in a room; and it's being illuminated with intensity I from one side.
The room temperature (T_{environment}) is set to be constant.
[ latex ]
\Delta Q = Q_{in} - Q_{out}\\
= I\Delta t A - P_{out}\Delta t + P_{in}\Delta t\\
= I\Delta t A - \sigma A T^{4} \Delta t + \sigma A T^{4}_{environment} \Delta t\\
\Delta Q = cm\Delta T\\
\frac{\Delta T}{\Delta t} = I\Delta t \pi r^2 - \sigma 4 \pi r^2 T^{4} \Delta t + \sigma 4 \pi r^2 T^{4}_{environment} \Delta t
[ /latex ]
Hence we have a differential equation in the form of
!
[ latex ]
\frac{dT(t)}{dt} = a - b*T^4(t)
[ /latex ]
!
I have tried different methods... But none of them worked... (And wolframalpha can only solve for t :-( )
Thanks for your consideration ;-)
atta_bo-y