I have been researching for a while looking for what to include more than just deriving the equation of Newton’s law of cooling and a website gave me inspiration by stating what if the temperature of the environment is changing (± C). In conclusion, I wasn’t sure if that person carried out the mathematical process in the right way and the only place I could ask for accurate information is here so please let me know what you think about the mathematical process he went for and if it is right or wrong.
Thanks a lot.
Website: http://calculus7.com/sitebuildercontent/sitebuilderfiles/nwtnslawcling1.pdf
Assuming that you refer to the equation $$y'(t)=\frac{3}{20} \left(55+10 \sin \left(\frac{\pi }{12}t\right)-y(t)\right)$$ we can integrate it. $$y(t)=55+C \,e^{-3 t/20}+\frac{15 \left(162 \sin \left(\frac{\pi }{12}t\right)-90 \pi \cos \left(\frac{\pi }{12}t\right)\right)}{243+75 \pi ^2}$$ Using the initial condition $y(0)=45$, we have $$C=\frac{450 \pi }{81+25 \pi ^2}-10$$