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I am interested to know whether a theory exists that allows to answer the following sort of question.

Does a solution of initial value problem of second order differential equation is infinitely differentiable on the set of positive real numbers?

For example,

1) the solution of {y''=y^2, y(0)=1, y'(0)=0} is a function y(t) that goes to infinity as t approaches 2.9744 ... and thus is not infinitely differentiable.

2) {y''=-1/y, y(0)=1,y'(0)=0} is not infinitely differentiable as well, since first order derivative goes to infinity as t approaches 1.25...

3) on the contrary, for {y''=1/y, y(0)=1,y'(0)=0}, y(t) is infinitely differentiable on the set of reals.

So, is there any theory which helps to answer such sort of questions without explicitely solving an equation or system of equations?

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# ? - infinitely differentiable solutions to initial value problems

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