# How can I solve the following ode

$$C\frac{dT}{dt}=-\sigma T(t)^{4}+(1-\alpha)Q$$ I need help to solve the above pde where $C,\alpha ,Q$ are constants, I'm really unsure on how to even start to solve it

• It looks like an ODE to me. I presume $t(t)$ is a typo? – Yuriy S Mar 14 '18 at 18:34
• Yes sorry that is a typo thanks – Gibberish Mar 14 '18 at 18:35
• Is $*$ multiplication? If so you should write $\sigma T(t)^4$ instead. If it is not you need to say so. – AzJ Mar 14 '18 at 18:36
• It is multiplication not convolutions – Gibberish Mar 14 '18 at 18:38
• $\sigma$ is a constant as well? You can separate variables, but the integral is gross – operatorerror Mar 14 '18 at 18:48

## 1 Answer

As you have written it, your problem is an ODE not a PDE as $T$ is a function of only one variable. As you only have one function and everything else is a constant you can solve using separation of variables.

For example let $p=(1-\alpha)Q$. We can can separate variables as

\begin{align} \frac{C}{-\sigma T^4+p} \frac{d T}{dt}=1 \end{align} The tricky part is integration of the left side. Also see

The second link we be more helpful as the it shows how to express the solution as an integral. I beilive this form is going to be part of the expected answer as this is a very diffcult integral.

• should $d$ be $C$? – Gibberish Mar 14 '18 at 18:56
• yes, my mistake – AzJ Mar 14 '18 at 18:57
• Please accept my answer, if you found it satisfactory. – AzJ Mar 14 '18 at 19:16