# Preface

This section shows application of Picard's iteration procedure to construct an integrating factor as an analytic function. variable. coordinates.

# Approximation of integrating factors

Consider the initial value problem

$y' = f(x,y) , \qquad y(x_0 ) = y_0 ,$
that we rewrite in differential form
${\text d}y - f(x,y)\,{\text d}x =0 .$
This equation may be solved upon finding an integrating factor μ(x,y) so that $$\mu\,{\text d}y - \mu\,f(x,y)\,{\text d}x = {\text d}\psi$$ is total differential of some function (not unique) ψ(x,y), called the potential function for the given differential equation $$y' = f(x,y) .$$ Its solution is obtained inplicitly given by the relationship ψ(x,y) = C, where C is an arbitrary constant. It is well known that the equation $$y' = f(x,y)$$ admits an integrating factor μ if and only if μ satisfies the first order differential equation
$- \frac{\partial}{\partial y} \left( \mu\,f \right) = \frac{\partial}{\partial x} \left( \mu \right) .$
Unfortunately, this equation os usually difficult to solve and leads to find integrator factors in special forms. Instead, we try to approximate an integrating factor using Picard's iteration procedure.

We rewrite the partial differential equation for μ in the following form:

$\frac{\partial \mu}{\partial x} = - f\,\frac{\partial \mu}{\partial y} - \mu\, \frac{\partial f}{\partial y} ,$
which is linear in μ. If we fix y, we get an equation in x to which we apply Picard's iterations:
\begin{align*} \mu_0 &= 1 \qquad (\mbox{or any number}) \\ \mu_{n+1} (x,y) &= \mu_0 - \int_0^x \left[ \frac{\partial \mu_n}{\partial y}\, (t,y) + \mu_n (t,y)\, \frac{\partial f}{\partial y} (t,y) \right] {\text d}t . \end{align*}
This sequence μn will converge to required integrating factor subject that all functions satisfy appropriate continuity conditions. Although Picard's iterations is in general hard to perform, we may try to find μ in the form:
$\mu (x,y) = \sum_{m\ge 0} \sum_{n\ge 0} a_{mn} x^m y^n ,$
Substitution of the latter into $$\mu_x = \mu_y \,f - \mu\,f_y$$ yields a recurrence relation (underdetermined) for the coefficients.

Example: Consider undamped Duffing equation
$\ddot{x} + 2\eta\, \dot{x} + x + \varepsilon \,x^3 = 0 .$
If we put $$x' = 1/y ,$$ the Duffing equation becomes ■
Example: Consider the van der Pol equation for x(t)
$x'' + \varepsilon \left( x^2 -1 \right) x' + x =0 .$
If we put $$x' = 1/y ,$$ the van der Pol equation becomes
$y' (x) = x\, y^3 - \varepsilon \left( 1 - x^2 \right) y^2 .$
1. Roman-Miller, L. and Smith, G.H., Analytic solutions of first-order nonlinear differential equations, International Journal of Mathematical Education in Science and Technology, 2010, Vol. 41, No. 5, pp. 649--665; https://doi.org/10.1080/00207391003675182