Algebrical equation

can you help me please to resolve this question.
I(x,t) is the solution of an variational solution system.
To find I(x,t) we use

for (real t=0;t<T;t+=dt) {

After this, how we can resolve the equation I(x,t)=0.5 for all t: we denote the solution x=x(t), and after this we calculate c(t)=x’(t) for all t.

Thank’s in advance .