# 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.
• So it would be correct if I renamed the constant $-c$ to $c_1$? – user707991 Sep 26 '19 at 23:08