Given the differential equation

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> $\frac{dx}{dt} = \lambda x$
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> 
> 

We have that general solution of this equation is:

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> $x(t) = \pm e^{C}e^{\lambda t}$, for $C \in \mathbb{R}$
> 
> 
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I'm having difficulty fully understanding the $\pm$ symbol in this solution. I can see that for any C, the function $x(t) = e^{C}e^{\lambda t}$ solves the differential equation and the equation $x(t) = - e^{C}e^{\lambda t}$ also solves the differential equation. However my question is, for a given function $x(t)$ that solves this differential equation, do you have to 'fix' either a '+' or a '-' or would a function that, for some $C \in \mathbb{R}$, takes the value $x(t) = e^{C}e^{\lambda t}$ for some values of $\mathbb{R}$ and takes the value $x(t) = - e^{C}e^{\lambda t}$ for all other values of $\mathbb{R}$ also solve the equation?