finding another particular solution of linear ODE

Consider the homogeneous (http://planetmath.org/HomogeneousLinearDifferentialEquation) second-order linear ordinary differential equation

$\displaystyle y^{\prime\prime}\!+\!P(x)y^{\prime}\!+\!Q(x)y\;=\;0.$

(1)

If one knows one particular solution (http://planetmath.org/SolutionsOfOrdinaryDifferentialEquation) $y=y_{1}(x)\not\equiv 0$ of (1), it’s possible to derive from it via two quadratures another solution $y_{2}(x)$ , linearly independent on $y_{1}(x)$ ; thus one can write the general solution

$y\;=\;C_{1}y_{1}(x)\!+\!C_{2}y_{2}(x)$

of that homogeneous differential equation.

We will now show the derivation procedure.

We put

$\displaystyle y\;=\;uv$

(2)

which renders (1) to

$\displaystyle(v^{\prime\prime}+Pv^{\prime}+Qv)u+(2v^{\prime}+Pv)u^{\prime}+u^{% \prime\prime}v\;=\;0.$

(3)

Here one can choose $v:=y_{1}(x)$ , whence the first addend vanishes, and (3) gets the form

$\displaystyle(2y_{1}^{\prime}+Py_{1})u^{\prime}+y_{1}u^{\prime\prime}\;=\;0.$

(4)

This equation may be written as $\frac{u^{\prime\prime}}{u^{\prime}}=-2\frac{y_{1}^{\prime}}{y_{1}}-P$ , which is integrated to

$\ln\left|\frac{du}{dx}\right|\;=\;\ln\frac{1}{y_{1}^{2}}-\int P\,dx+\mbox{% constant},$

i.e.

$\frac{du}{dx}\;=\;\frac{C}{y_{1}^{2}}e^{-\int P\,dx}.$

A new integration results from this the general solution of (4):

$u\;=\;C\int\frac{e^{-\int P\,dx}}{y_{1}^{2}}\,dx+C^{\prime}.$

Thus by (2), we have obtained the wanted other solution

$y_{2}(x)\;=\;y_{1}(x)\int\frac{e^{-\int P\,dx}}{y_{1}^{2}}\,dx$

which is clearly linearly independent on y_1(x).

Consequently, we can express the general solution of the differential equation (1) as

$y\;=\;y_{1}(x)u\;=\;C_{1}y_{1}(x)+C_{2}y_{1}(x)\int\frac{e^{-\int P\,dx}}{y_{1% }^{2}}\,dx,$

where $C_{1}$ and $C_{2}$ are arbitrary constants.

Remark. The substitution

$y\;:=\;e^{-\frac{1}{2}\int P(x)\,dx}u$

converts the equation (1) into the form

$\frac{d^{2}u}{dx^{2}}+(Q-\frac{P^{2}}{4}-\frac{P^{\prime}}{2})u\;=\;0$

not containing the derivative $\frac{du}{dx}$ .

References

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

Title	finding another particular solution of linear ODE
Canonical name	FindingAnotherParticularSolutionOfLinearODE
Date of creation	2014-02-28 14:31:42
Last modified on	2014-02-28 14:31:42
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	10
Author	pahio (2872)
Entry type	Algorithm
Classification	msc 34A05