linear differential equation of first order

An ordinary linear differential equation of first order has the form

dydx+P(x)y=Q(x), (1)

where y means the unknown function, P and Q are two known continuous functionsMathworldPlanetmathPlanetmath.

For finding the solution of (1), we may seek a function y which is productPlanetmathPlanetmath 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 derivativePlanetmathPlanetmathdydx=udvdx+vdudx  in (1), getting  udvdx+vdudx+Puv=Q,  or

u(dvdx+Pv)+vdudx=Q. (3)

If we chose the function v such that

dvdx+Pv= 0,

this condition may be written


Integrating here both sides gives  lnv=-P𝑑x  or


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

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






So we have obtained the solution

y=e-P(x)𝑑x[C+Q(x)eP(x)𝑑x𝑑x] (4)

of the given differential equationMathworldPlanetmath (1).

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


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 DerivationMathworldPlanetmath
Classification msc 34A30
Synonym linear ordinary differential equation of first order
Related topic SeparationOfVariables