How can I solve the following ode

Question

$$C\frac{dT}{dt}=-\sigma T(t)^{4}+(1-\alpha)Q$$ 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

Answer 1

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

• http://tutorial.math.lamar.edu/Classes/DE/Separable.aspx
• https://en.wikipedia.org/wiki/Ordinary_differential_equation#Summary_of_exact_solutions

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.