1.1 Algorithm

This gives an overview on solving first order ode where \(y^{\prime }\) enters the ode as nonlinear. Examples are \(x\left ( y^{\prime }\right ) ^{2}+yy^{\prime }+x=0\) or \(2y^{\prime }x-y+\ln y^{\prime }=0\) and so on. Four general cases exist and these are summarized in the flow chart at the end of this section. Two of these cases are called the Clairaut ode and the d’Alembert ode. Following the flow chart, a number of examples are solved.

Given the ode \(F\left ( x,y,y^{\prime }\right ) =0\), we start by writing \(y^{\prime }=p\) which results in

\[ F\left ( x,y,p\right ) =0 \]

This is the top level algorithm

Figure 1: Algorithm for first order nonlinear in derivative

Algorithm below is Clairaut dAlembert solver algorithm

Figure 2: Algorithm for clairaut solver

The following is the general flow chart for solving nonlinear first order ode using d’Alembert and Clairaut methods.

Figure 3: Algorithm for solving first order ode with nonlinear \(y'\)