Determining the maximum intervall of existence Can somebody explain to me the concept of an intervall of existance/interval of validity? Is it basically the domain of my differential equation? 
I tried to look it up and I came across this site
In the article it says:

Consider the following IVP 
$$y'+p(t)y=g(t); \space \space y(t_0)=y_0$$
...if the interval in the theorem is the largest possible interval on
  which $p(t)$ and $g(t)$ are continuous then the interval is the interval
  of validity for the solution. This means, that for linear first order
  differential equations, we won't need to actually solve the
  differential equation in order to find the interval of validity.
  Notice as well that the interval of validity will depend only
  partially on the initial condition. The interval must contain $t_0$, but
  the value of $y_0$, has no effect on the interval of validity.

Suppose I have the following linear first order differential equation:
$$tx'(t)+3x(t)=-\frac{1}{t^2+1}, \space \space \space \space x(1)=\frac{\pi}{4}$$
$$\iff x'(t)+\frac{3x}{t}x(t)=-\frac{1}{t^3+t}$$
So in this case $p(t)=\frac{3x}{t}$ and $g(t)=-\frac{1}{t^3+t}$
For $p(t)$ I get the intervall $(-\infty,0) \cup(0,\infty)$
For $g(t)$ I get the intervall $(-\infty,0) \cup(0,\infty)$
Since my intervall must contain $1$ the maximum intervall of existance is $(0,\infty)$
Is this correct?
How can I do this for nonlinear differential equations like:
$x'(t)=e^{x(t)}\cos(t); \space x(0)=x_0$
$x'(t)=c\cdot x(t) \cdot (1-x(t)); \space x(0)=x_0$
Is the (maximum) intervall of existance only something that makes sense when talking about initial value problems?
 A: To summarize some of the comments: The existence and uniqueness theorems for solutions of the functional equation $\dot x=f(t,x)$ say:


*

*Peano: If $f$ is continuous on some domain $D$, then every IVP with initial value in $D$ has a local solution (not necessarily unique).

*Picard-Lindelöf: If $f$ is continuous in $D$ and has a Lipschitz constant in direction $x$ for all of $D$, then local solutions exist and are unique.

*Cauchy-???: If $f$ is continuous in $D$ and is continuously differentiable in direction $x$, then local solutions exist and are unique.
The maximal solution statement essentially is that the solutions can be continued to the boundary of $D$ (or infinity, if that part of the boundary does not exist).
The Cauchy theorem is the most practical. Locally it is a consequence of the Picard-Lindelöf theorem, globally it is less restrictive (for differentiable ODE functions). Differentiability is often easier to check than bounding the difference quotient. 
Linear ODE with continuous coefficients (almost) trivially satisfy the Lipschitz condition of Picard-Lindelöf. Since there is no boundary in $x$ direction, the domain of the maximal solution is determined by the continuity intervals of the coefficients.
In the manual solution of ODE, there is no difference of local and global solutions. The expression for the local solution is also the expression for the extension. In some cases, the expression for the solution contains singularities that may or may not be connected to singularities of the ODE function. The domain of the maximal solution ends at such a singularity, even if the expression gives values beyond.
