# method of integrating factors

The method of integrating factors is in principle a means for solving ordinary differential equations^{} of first . It has not great practical significance, but is theoretically important.

Let us consider a differential equation solved for the derivative^{} ${y}^{\prime}$ of the unknown function and write the equation in the form

$X(x,y)dx+Y(x,y)dy=\mathrm{\hspace{0.33em}0}.$ | (1) |

We assume that the functions $X$ and $Y$ have continuous^{} partial derivatives^{} in a region $R$ of ${\mathbb{R}}^{2}$.

If there is a solution of (1) which may be expressed in the form

$$f(x,y)=C$$ |

with $f$ having continuous partial derivatives in $R$ and with $C$ an arbitrary constant, then it’s not difficult to see that such an $f$ satisfies the linear partial differential equation

$X{\displaystyle \frac{\partial f}{\partial y}}-Y{\displaystyle \frac{\partial f}{\partial x}}=\mathrm{\hspace{0.33em}0}.$ | (2) |

Conversely, every non-constant solution $f$ of (2) gives also a solution $f(x,y)=C$ of (1). Thus, solving (1) and solving (2) are equivalent^{} (http://planetmath.org/Equivalent3) tasks.

It’s straightforward to show that if ${f}_{0}(x,y)$ is a non-constant solution of the equation (2), then all solutions of this equation are $F({f}_{0}(x,y))$ where $F$ is a freely chosen function with (mostly) continuous derivative.

The connection of the equations (1) and (2) may be presented also in another form. Suppose that $f(x,y)=C$ is any solution of (1). Then (2) implies the proportion equation

$$\frac{{f}_{x}^{\prime}}{X}=\frac{{f}_{y}^{\prime}}{Y}.$$ |

If we denote the common value of these two ratios by $\mu (x,y)=\mu $, then we have

$${f}_{x}^{\prime}=\mu X,{f}_{y}^{\prime}=\mu Y.$$ |

This gives to the differential^{} of the function $f$ the expression

$$df(x,y)=\mu (x,y)(X(x,y)dx+Y(x,y)dy).$$ |

We see that $\mu (x,y)$ is the integrating factor or Euler multiplicator of the given differential equation (1), i.e. the left hand side of (1) turns, when multiplied by $\mu (x,y)$, to an exact differential (http://planetmath.org/ExactDifferentialForm).

Conversely, any integrating factor $\mu $ of (1), i.e. such that $\mu Xdx+\mu Ydy$ is the differential of some function $f$, is easily seen to determine the solutions of the form $f(x,y)=C$ of (1). Altogether, solving the differential equation (1) is equivalent with finding an integrating factor of the equation.

When an integrating factor $\mu $ of (1) is available, the solution function $f$ can be gotten from the line integral

$$f(x,y)=:{\int}_{{P}_{0}}^{P}[\mu (x,y)X(x,y)dx+\mu (x,y)Y(x,y)dy]$$ |

along any curve $\gamma $ connecting an arbitrarily chosen point ${P}_{0}=({x}_{0},{y}_{0})$ and the point $P=(x,y)$ in the region $R$.

Note. In general, it’s very hard to find a suitable integrating factor. One special case where such can be found, is that $X$ and $Y$ are homogeneous functions of same degree (http://planetmath.org/HomogeneousFunction): then the expression $\frac{1}{xX+yY}$ is an integrating factor.

Example. In the differential equation

$$({x}^{4}+{y}^{4})dx-x{y}^{3}dy=\mathrm{\hspace{0.33em}0}$$ |

we see that $X=:{x}^{4}+{y}^{4}$ and $Y=:-x{y}^{3}$ both define a homogeneous function of degree (http://planetmath.org/HomogeneousFunction) 4. Thus we have the integrating factor $\mu =:{\displaystyle \frac{1}{{x}^{5}+x{y}^{4}-x{y}^{4}}}={\displaystyle \frac{1}{{x}^{5}}}$, and the left hand side of the equation

$$\left(\frac{1}{x}+\frac{{y}^{4}}{{x}^{5}}\right)dx-\frac{{y}^{3}}{{x}^{4}}dy=\mathrm{\hspace{0.33em}0}$$ |

is an exact differential. We can integrate it along the broken line, first from $(1,\mathrm{\hspace{0.17em}0})$ to $(x,\mathrm{\hspace{0.17em}0})$ and then still to $(x,y)$, obtaining

$$f(x,y)=:{\int}_{1}^{x}(\frac{1}{x}+\frac{{0}^{4}}{{x}^{5}})dx-{\int}_{0}^{y}\frac{{y}^{3}dy}{{x}^{4}}=\mathrm{ln}|x|-\frac{{y}^{4}}{4{x}^{4}}.$$ |

So the general solution of the given differential equation is

$$\mathrm{ln}|x|-\frac{{y}^{4}}{4{x}^{4}}=C.$$ |

## References

- 1 E. Lindelöf: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title | method of integrating factors |
---|---|

Canonical name | MethodOfIntegratingFactors |

Date of creation | 2013-03-22 16:31:48 |

Last modified on | 2013-03-22 16:31:48 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 21 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 35-00 |

Classification | msc 34-00 |

Related topic | ErnstLindelof |

Defines | integrating factor |

Defines | Euler multiplicator |