Introduction to ODEs

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Here we want to solve a Cauchy problem as

$\{\begin{array}{c}{u}^{\prime }(t)=f(u,t)\\ u(0)=u_0\end{array}$.

where $f$ is a given real-valued function of two real variables. In order to select the solution among the infinite family of solutions coming from the integration of $u^{\prime }(t)$ we need to specify the initial data $u(0)=u_0$.

In other words, we seek the unique solution $u(t)$ of the above-mentioned Cauchy problem for $t>0$ and subject to the condition $u(0)=u_0$.

Since it is often impossible to construct an analytical solution, we look for an approximate solution of the problem which will converge, in some sense to be specified later, to the exact solution.