# linear differential equation of first order

An ordinary linear differential equation of first order has the form

$\frac{dy}{dx}}+P(x)y=Q(x),$ | (1) |

where $y$ means the unknown function, $P$ and $Q$ are two known continuous functions^{}.

For finding the solution of (1), we may seek a function $y$ which is product^{} of two functions:

$y(x)=u(x)v(x)$ | (2) |

One of these two can be chosen freely; the other is determined according to (1).

We substitute (2) and the derivative^{} $\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$ in (1), getting
$u\frac{dv}{dx}+v\frac{du}{dx}+Puv=Q$, or

$u\left({\displaystyle \frac{dv}{dx}}+Pv\right)+v{\displaystyle \frac{du}{dx}}=Q.$ | (3) |

If we chose the function $v$ such that

$$\frac{dv}{dx}+Pv=\mathrm{\hspace{0.33em}0},$$ |

this condition may be written

$$\frac{dv}{v}=-Pdx.$$ |

Integrating here both sides gives $\mathrm{ln}v=-\int P\mathit{d}x$ or

$$v={e}^{-{\scriptscriptstyle \int P\mathit{d}x}},$$ |

where the exponent means an arbitrary antiderivative of $-P$. Naturally, $v(x)\ne 0$.

Considering the chosen property of $v$ in (3), this equation can be written

$$v\frac{du}{dx}=Q,$$ |

i.e.

$$\frac{du}{dx}=\frac{Q(x)}{v(x)},$$ |

whence

$$u=\int \frac{Q(x)}{v(x)}\mathit{d}x+C=C+\int Q{e}^{{\scriptscriptstyle \int P\mathit{d}x}}\mathit{d}x.$$ |

So we have obtained the solution

$y={e}^{-{\scriptscriptstyle \int P(x)\mathit{d}x}}\left[C+{\displaystyle \int Q(x){e}^{{\scriptscriptstyle \int P(x)\mathit{d}x}}\mathit{d}x}\right]$ | (4) |

of the given differential equation^{} (1).

The result (4) presents the general solution of (1), since the arbitrary $C$ may be always chosen so that any given initial condition^{}

$$y={y}_{0}\mathit{\hspace{1em}}\mathrm{when}\mathit{\hspace{1em}}x={x}_{0}$$ |

is fulfilled.

Title | linear differential equation of first order |
---|---|

Canonical name | LinearDifferentialEquationOfFirstOrder |

Date of creation | 2013-03-22 16:32:09 |

Last modified on | 2013-03-22 16:32:09 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Derivation^{} |

Classification | msc 34A30 |

Synonym | linear ordinary differential equation of first order |

Related topic | SeparationOfVariables |