The differential equation y'[t]^2+y''[t]==0 with initial conditions y'[1] ==1 and y[1] == 0 contains a pole at t = 0 however, is there a way to find out that a given differential equation with provided initial conditions has poles and what the radius of convergence for the Taylor series representing the solution of the ordinary differential equation may be?
y'[t]^2+y''[t]==0
y'[1] ==1
y[1] == 0
t = 0
The original poster isn't participating. I add the question "what do you mean by pole" because typical (mechanical wave texts) do not use the term and plots are non-polar if we discount sinusoidal behavior as poles (usually referred to in those terms not polar terms imho).
The time for $y' \rightarrow \infty$ is given by $\int_{y'_0}^\infty dt/y''$:
Solve[y'[t]^2 + y''[t] == 0, y''[t]] dt = Integrate[1/y''[t] /. First[%], {y'[t], 1, Infinity}] (* {{(y''[t] -> -y'[t]^2}} -1 *)
