Solution of homogeneous linear equation containing parameter How do you find a solution of this differential equation on both of its intervals of existence?
Is this correct?
$x'=\frac{k}{k+t}x \implies x'-\frac{k}{k+t}x=0$
$\implies (|k+t|^{-k}x)'=0  \implies x(t)=c|k+t|^{k}$
x(t)=\begin{cases} 
      c(k+t)^{k} & k>-t \\
       -c(k+t)^k &  k<-t \\
       \end{cases}

This is mostly right, but there is one issue. Because of the singularity at $t = -k$, you have two separate problems on $t > -k$ and $t < -k$. In terms of your solution, this would mean you can have two different constants $c$ on these two domains.
