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Hi,
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?
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?