Differential Equations without Analytical Solutions

Solution 1:

Take the initial value problem $$y'=\cases{x\bigl(1+2\log|x|\bigr)\quad &$(x\ne0)$ \cr 0&$(x=0)$\cr}\ ,\qquad y(0)=0\ .$$ This example obviously fulfills the assumptions of the existence and uniqueness theorem, so there is exactly one solution. As is easily checked this solution is given by $$y(x)=\cases{x^2\>\log|x|\quad&$(x\ne0)$\cr 0&$(x=0)$\cr}\ .$$ This function is not analytic in any neighborhood of $x=0$.

Solution 2:

There's something worse than having no analytical solution. Pour-El and Richards found an ordinary differential equation $\phi'(t)=F(t,\phi(t))$ with $F$ computable and no computable solution. A reference is Marian Boykan Pour-El and Ian Richards, A computable ordinary differential equation which possesses no computable solution, Ann. Math. Logic 17 (1979), no. 1-2, 61–90, MR0552416 (81k:03064).