Preface
This section shows application of Picard's iteration procedure to construct an integrating factor as an analytic function. variable. coordinates.
Return to computing page for the second course APMA0340
Return to Mathematica tutorial for the second course APMA0340
Return to the main page for the course APMA0330
Return to the main page for the course APMA0340
Return to Part II of the course APMA0330
Glossary
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
\[
y' (x) = x\, y^3 - \varepsilon \left( 1 - x^2 \right) y^2 .
\]
■ - 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
Return to Mathematica page
Return to the main page (APMA0330)
Return to the Part 1 (Plotting)
Return to the Part 2 (First Order ODEs)
Return to the Part 3 (Numerical Methods)
Return to the Part 4 (Second and Higher Order ODEs)
Return to the Part 5 (Series and Recurrences)
Return to the Part 6 (Laplace Transform)
Return to the Part 7 (Boundary Value Problems)