Rewrite this problem in standard form, as a system of first-order ODEs

Question

Let us consider the following initial value problem for a second-order equation:

$$y''= xy'-1/(1+y) \qquad\text{with}\quad y(1)=1, \;\; y'(1)=1$$

I need to rewrite this problem in standard form, as a system of first-order ODEs.

Any ideas?

Answer 1

$$y''= xy'-1/(1+y) $$$$\qquad\text{with}\quad y(1)=1, \;\; y'(1)=1$$ Substitute: $$u=y, v=y'$$ You have now a system of two differential equations of first order: $$ \begin{cases} u'=v \\ v'=xv-\dfrac {1}{1+u} \end{cases} $$ Where the initial conditions are : $$ \begin{cases} u(1)=y(1)=1 \\ v(1)=y'(1)=1 \end{cases} $$