# exact differential equation

Let $R$ be a region in ${\mathbb{R}}^{2}$ and let the functions $X:R\to \mathbb{R}$, $Y:R\to \mathbb{R}$ have continuous^{} partial derivatives^{} in $R$. The first order differential equation^{}

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

or

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

is called an exact differential equation, if the condition

$\frac{\partial X}{\partial y}}={\displaystyle \frac{\partial Y}{\partial x}$ | (2) |

is true in $R$.

By (2), the left hand side of (1) is the total differential^{} of a function, there is a function $f:R\to \mathbb{R}$ such that the equation (1) reads

$$df(x,y)=\mathrm{\hspace{0.33em}0},$$ |

whence its general integral is

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

The solution function $f$ can be calculated as the line integral

$f(x,y):={\displaystyle {\int}_{{P}_{0}}^{P}}[X(x,y)dx+Y(x,y)dy]$ | (3) |

along any curve $\gamma $ connecting an arbitrarily chosen point ${P}_{0}=({x}_{0},{y}_{0})$ and the point $P=(x,y)$ in the region $R$ (the integrating factor^{} is now $\equiv 1$).

Example. Solve the differential equation

$$\frac{2x}{{y}^{3}}dx+\frac{{y}^{2}-3{x}^{2}}{{y}^{4}}dy=\mathrm{\hspace{0.33em}0}.$$ |

This equation is exact, since

$$\frac{\partial}{\partial y}\frac{2x}{{y}^{3}}=-\frac{6x}{{y}^{4}}=\frac{\partial}{\partial x}\frac{{y}^{2}-3{x}^{2}}{{y}^{4}}.$$ |

If we use as the integrating way the broken line from $(0,\mathrm{\hspace{0.17em}1})$ to $(x,\mathrm{\hspace{0.17em}1})$ and from this to $(x,y)$, the integral (3) is simply

$${\int}_{0}^{x}\frac{2x}{{1}^{3}}\mathit{d}x+{\int}_{1}^{y}\frac{{y}^{2}-3{x}^{2}}{{y}^{4}}\mathit{d}y=\frac{{x}^{2}}{{y}^{3}}-\frac{1}{y}+1={x}^{2}-\frac{1}{y}+\frac{{x}^{2}}{{y}^{3}}+1-{x}^{2}=\frac{{x}^{2}}{{y}^{3}}-\frac{1}{y}+1.$$ |

Thus we have the general integral

$$\frac{{x}^{2}}{{y}^{3}}-\frac{1}{y}=C$$ |

of the given differential equation.

Title | exact differential equation |
---|---|

Canonical name | ExactDifferentialEquation |

Date of creation | 2013-03-22 18:06:17 |

Last modified on | 2013-03-22 18:06:17 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 34A05 |

Related topic | HarmonicConjugateFunction |

Related topic | Differential |

Related topic | TotalDifferential |

Defines | exact differential equation |