When reading the next question about Differential Equations and their application in Finance (in this scenario they are modelling Annuities; Differential Equation) The Differential equation is:
$$\frac{dM}{dt} = rM - p$$
And the solution may have been derived as
$$dM = r(M-\frac{p}{r})dt$$
$$\frac{dM}{M - \frac{p}{r}} = r dt$$
After the integrals:
$$ln(M - \frac{p}{r}) = rt + c$$
$$M = e^{rt+c} + \frac{p}{r}$$
$$M = Ce^{rt} + \frac{p}{r}$$
I tried to solve that equation before watching their solution, but forgot to divide everything by r. So I did:
$$dM = (rM-p)dt$$
$$\frac{dM}{rM-p} = dt$$
After the integration
$$ln(rM - p) = t + c$$
$$M = \frac{e^{t+c} + p}{r}$$
$$M = \frac{Ce^{t} + p}{r}$$
I think that this equations are certainly not the same.
$$Ce^{rt} + \frac{p}{r} \neq \frac{Ce^{t} + p}{r}$$
Can you explain me what's wrong in the second process? (Or if they are the same, how is the $r$ embedded in the constant $C$?)
In the second approach, in fact $\int \frac{1}{rM-p}dM = \frac{1}{r}\ln(rM-p)$. So $\frac{1}{r}\ln(rM-p) = t + c \implies M = \frac{e^{rt + rc} + p}{r} = Ce^{rt} + \frac{p}{r}$, where I defined $C = \frac{e^{rc}}{r}$. Since r and c are constants this is possible.