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I attempted to solve the following first-order linear ODE: $$ \frac{dn}{dt} = rn(t)+m $$

This is what I managed to come up with: \begin{align} \int \frac{1}{rn(t)+m}dn & = \int 1dt \\ \frac{\ln(rn(t)+m)}{r} & = t+C \\ rn(t)+m & = e^{r(t+C)}\\ rn(t)+m & = e^{rt}e^{rC} \end{align}


$$ rn(0)+m = e^{rC} $$


\begin{align} rn(t)+m & = e^{rt}(rn(0)+m) \\ n(t) & = \frac{e^{rt}rn(0)+e^{rt}m-m}{r} \\ n(t) & = \frac{e^{rt}rn(0)+(e^{rt}-1)m}{r} \\ n(t) & = e^{rt}n(0)-\frac{m}{r}(1-e^{rt}) \end{align}

This solution should be correct based on the book where I got the ODE from. (Otto, S.P.; Day, T., 2007. A Biologist's Guide to Mathematical Modeling in Ecology and Evolution. Princeton University Press. (p. 200.: Recipe 6.3)) My purpose was to work the solution out by a separation of variables by myself.
But when I give $\frac{dn}{dt}=rn(t)+m$ to WolframAlpha, its solution is the following: $$ n(t) = c_1e^{rt}-\frac{m}{r} $$

Where does this come from? And how do these two solutions correspond/relate to each other? I've tried but could not work it out.

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They are the same, we will find $c_{1}.$

Let $t=0$ in the WolframAlpha solution, then $$n(0)=c_{1}-\frac{m}{r}$$

$$c_{1}=n(0)+\frac{m}{r}$$ and thus we have $$n(t)=c_{1}e^{rt}-\frac{m}{r}$$ $$=e^{rt}(n(0)+\frac{m}{r})-\frac{m}{r}$$ $$=e^{rt}n(0)-\frac{m}{r}(1-e^{rt}).$$


You showed that $rn(t)+m=e^{rt}e^{rC}$ or $n(t)=\frac{e^{rC}}{r}e^{rt}-\frac{m}{r}.$ Then since $\frac{e^{rC}}{r}$ is just another constant, we can let $\frac{e^{rC}}{r}=c_{1}$ to obtain the WolframAlpha solution $n(t)=c_{1}e^{rt}-\frac{m}{r}.$ The constant is fixed by the boundary condition at $t=0$.

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