I have a differential equations question:
Given the differential equation $$t(t+1) \frac{ds}{dt} = sln(s)$$ it can be trivially solved to be $$ln ln s = ln(t) - ln(t+1) +C$$
As known, C is the constant of integration.
I first tried to solve the equation for C and got a weird value when given the IVP parameter $S(3) = 2$. To try and see where I went wrong, I plugged the differential equation into the math software Maple and it spat out the answer $$s(t) = e^\frac{t*c}{t+1}$$ (it might be tricky to see, but the numerator of the power on top of the Euler's number is $t*C$).
My understanding is that Maple added the natural log to C such that the final equation becomes $$ln ln s = ln(t) - ln(t+1) + ln(C)$$ Where simple algebraic manipulation can achieve the result.
My question then is, how can Maple just do that? Is it okay to algebraically do whatever we want to the constant of integration so that we can make the equation solve easier? Thanks for your help in advanced - I really appreciate it!
P.S. Would it be too much to ask to keep the answer as simple as possible? Thanks so much!