text: For cylindrical vessel, there is an exact solution. Though Frank-Kamentskii used numerical integration assuming there is no explicit solution, Paul L. Chambré provided an exact solution in 1952. H. Lemke also solved provided a solution in a somewhat different form in 1913. Here , then
  If the transformations  and  are introduced
  The general solution is . But  from the symmetry condition at the centre. Writing back in original variable, the equation reads,
  But the original equation multiplied by  is
  Now subtracting the last two equation from one another leads to
  This equation is easy to solve because it involves only the derivatives, so letting  transforms the equation
  This is a Bernoulli differential equation of order , a type of Riccati equation. The solution is
  Integrating once again, we have  where . We have used already one boundary condition, there is one more boundary condition left, but with two constants . It turns out  and  are related to each other, which is obtained by substituting the above solution into the starting equation we arrive at . Therefore, the solution is
  Now if we use the other boundary condition , we get an equation for  as . The maximum value of  for which solution is possible is when , so the critical Frank-Kamentskii parameter is . The system has no steady state( or explodes) for  and for , the system goes to a steady state with very slow reaction. The maximum temperature  occurs at 
  For each value of , we have two values of  since  is multi-valued. The maximum critical temperature is .