D'Alembert reduction process

The d'Alembert reduction method is a method from the theory of ordinary differential equations named after the mathematician and physicist Jean-Baptiste le Rond d'Alembert . It is used to trace a -th order linear differential equation with non-constant coefficients to a -th order linear differential equation with knowledge of a solution to the homogeneous problem . ${\displaystyle n}$ ${\displaystyle (n-1)}$

Roughly described, the following applies: To solve an (inhomogeneous) linear differential equation -th order , obtain a nontrivial solution of the associated homogeneous linear differential equation . Then the approach , i.e. the variation of the constants , leads to an inhomogeneous linear differential equation of the lower order for the original equation . ${\displaystyle n}$ ${\displaystyle L(y)=f}$ ${\displaystyle L(u)=0}$ ${\displaystyle y(x):=c(x)u(x)}$ ${\displaystyle L(y)=f}$ ${\displaystyle {\tilde {L}}(c')=f}$ ${\displaystyle n-1}$ ${\displaystyle c'(x)}$

Formulation of the sentence

Consider the -th order differential operator ${\displaystyle n}$

${\displaystyle L(v)(x):=\sum _{k=0}^{n}a_{k}(x)v^{(k)}(x)\ .}$

For this, let us solve the homogeneous linear differential equation ${\displaystyle u(x)}$

${\displaystyle L(u)=0}$

known. For

${\displaystyle y(x):=c(x)u(x)}$

then applies

${\displaystyle L(y)(x)=\sum _{j=0}^{n-1}\left[\sum _{k=j+1}^{n}{k \choose {j+1}}a_{k}(x)u^{(k-j-1)}(x)\right]c^{(j+1)}(x).}$

In other words: solves the inhomogeneous differential equation -th order if and only if ${\displaystyle y(x)}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {L}}(y)=f(x)}$

${\displaystyle z(x):=c'(x)}$

the inhomogeneous linear differential equation -th order ${\displaystyle (n-1)}$

${\displaystyle \sum _{j=0}^{n-1}\left[\sum _{k=j+1}^{n}{k \choose {j+1}}a_{k}(x)u^{(k-j-1)}(x)\right]z^{(j)}(x)=f(x)}$

solves.

proof

According to Leibniz's rule, the following applies

${\displaystyle (c\cdot u)^{(k)}(x)=\sum _{j=0}^{k}{k \choose j}c^{(j)}(x)u^{(k-j)}(x)\ ,}$

so

${\displaystyle \sum _{k=0}^{n}a_{k}(x)(c\cdot u)^{(k)}(x)=\sum _{k=0}^{n}\sum _{j=0}^{k}{\binom {k}{j}}a_{k}(x)c^{(j)}(x)u^{(k-j)}(x)=\sum _{j=0}^{n}\sum _{k=j}^{n}{\binom {k}{j}}a_{k}(x)u^{(k-j)}(x)c^{(j)}(x)\ .}$

The double sum indicates that the derivatives of are now added up. ${\displaystyle \textstyle \sum _{j=0}^{n}\sum _{k=j}^{n}{\binom {k}{j}}a_{k}(x)u^{(k-j)}(x)c^{(j)}(x)}$ ${\displaystyle c^{(j)}(x)}$

Now is according to the prerequisite and thus the 0th term in the sum is omitted , so that it follows ${\displaystyle \textstyle \sum _{k=0}^{n}{\binom {k}{0}}a_{k}(x)u^{(k)}(x)=L(u)=0}$ ${\displaystyle j}$

${\displaystyle L(y)=\sum _{k=0}^{n}a_{k}(x)(c\cdot u)^{(k)}(x)=\sum _{j=1}^{n}\left[\sum _{k=j}^{n}{\binom {k}{j}}a_{k}(x)u^{(k-j)}(x)\right]c^{(j)}(x)\ .}$

The index shift provides the result

${\displaystyle L(y)=\sum _{j=0}^{n-1}\left[\sum _{k=j+1}^{n}{\binom {k}{j+1}}a_{k}(x)u^{(k-j-1)}(x)\right]c^{(j+1)}(x)}$,

or using ${\displaystyle z(x)=c'(x)}$

${\displaystyle L(y)=\sum _{j=0}^{n-1}\left[\sum _{k=j+1}^{n}{\binom {k}{j+1}}a_{k}(x)u^{(k-j-1)}(x)\right]z^{(j)}(x)}$.

${\displaystyle \Box }$

example

The homogeneous linear differential equation of the 2nd order with constant coefficients is given

${\displaystyle u''(x)+4u'(x)+4u(x)=0}$.

A solution of the differential equation results from the characteristic equation with the double zero . With the help of the reduction method, the second linearly independent solution is found using the already known solution. With the approach of the variation of the constants follows ${\displaystyle \lambda ^{2}+4\lambda +4=0}$ ${\displaystyle \lambda _{1,2}=-2}$ ${\displaystyle u(x)=e^{-2x}}$

${\displaystyle y(x)=c(x)u(x)}$

and the given differential equation is represented as follows

${\displaystyle {\big (}c''(x)u(x)+2c'(x)u'(x)+c(x)u''(x){\big )}+4{\big (}c'(x)u(x)+c(x)u'(x){\big )}+4c(x)u(x)=0}$.

By rearranging the differential equation according to the derivatives of, we get ${\displaystyle c(x)}$

${\displaystyle u(x)c''(x)+{\big (}2u'(x)+4u(x){\big )}c'(x)+{\big (}u''(x)+4u'(x)+4u(x){\big )}c(x)=0}$.

The differential equation is expressed in the third term and is therefore not applicable. The differential equation is now ${\displaystyle u''(x)+4u'(x)+4u(x)=0}$

${\displaystyle u(x)c''(x)+\left(2u'(x)+4u(x)\right)c'(x)=0}$

and results with the already known solution for the second term , so that the differential equation is reduced to ${\displaystyle u(x)=e^{-2x}}$ ${\displaystyle 2u'(x)+4u(x)=-4e^{-2x}+4e^{-2x}=0}$

${\displaystyle u(x)c''(x)=0}$.

Since the exponential function represents and is therefore greater than zero everywhere, the condition for the second solution is the differential equation ${\displaystyle u(x)}$

${\displaystyle c''(x)=0}$.

By integrating twice, we get with the integration constants ${\displaystyle c_{1},c_{2}}$

${\displaystyle c(x)=c_{1}x+c_{2}}$.

The approach for the second solution of the differential equation thus results

${\displaystyle y(x)=(c_{1}x+c_{2})u(x)=c_{1}xu(x)+c_{2}u(x)}$.

Since the second term is only a scalar multiple of the first solution and is therefore linearly dependent, the second solution of the differential equation reads, omitting the constant of integration ${\displaystyle c_{2}u(x)}$

${\displaystyle y(x)=xu(x)=xe^{-2x}.}$

Finally, the Wronsky determinant can be used to prove the linear independence of the two solutions

${\displaystyle W(u,y)(x)={\begin{vmatrix}u&xu\\u'&u+xu'\end{vmatrix}}=u(u+xu')-xuu'=u^{2}=e^{-4x}\neq 0.}$

Special case: linear differential equation of the second order

Let the homogeneous linear differential equation of the second order be solved ${\displaystyle u(x)}$

${\displaystyle u''(x)+p(x)u'(x)+q(x)u(x)=0\ .}$

Then

${\displaystyle y(x):=c(x)u(x)}$

Solution of the (inhomogeneous) differential equation

${\displaystyle y''(x)+p(x)y'(x)+q(x)y(x)=f(x)}$

exactly when

${\displaystyle z(x):=c'(x)}$

the equation

${\displaystyle u(x)z'(x)+{\big (}p(x)u(x)+2u'(x){\big )}z(x)=f(x)}$

enough. This equation can be completely solved with the help of the variation of the constants .

proof

Let be the inhomogeneous linear differential equation

${\displaystyle y''(x)+p(x)y'(x)+q(x)y(x)=f(x)}$

given whose solution for the homogeneous differential equation is known. Then the solution of the (inhomogeneous) differential equation results using the approach of the variation of the constants by ${\displaystyle u(x)}$

${\displaystyle y(x)=c(x)u(x)}$,

where is any function. So is ${\displaystyle c(x)}$

${\displaystyle y'(x)=c'(x)u(x)+c(x)u'(x)}$

and

${\displaystyle y''(x)=c''(x)u(x)+2c'(x)u'(x)+c(x)u''(x)}$.

It follows

${\displaystyle {\big (}c''(x)u(x)+2c'(x)u'(x)+c(x)u''(x){\big )}+p(x){\big (}c'(x)u(x)+c(x)u'(x){\big )}+q(x)c(x)u(x)=f(x)}$

and by rearranging according to the derivatives of ${\displaystyle c(x)}$

${\displaystyle u(x)c''(x)+{\big (}p(x)u(x)+2u'(x){\big )}c'(x)+{\big (}u''(x)+p(x)u'(x)+q(x)u(x){\big )}c(x)=f(x)}$.

Since there is a solution to the homogeneous differential equation , the inhomogeneous differential equation can be reduced by this term and it applies ${\displaystyle u(x)}$ ${\displaystyle u''(x)+p(x)u'(x)+q(x)u(x)=0}$

${\displaystyle u(x)c''(x)+{\big (}p(x)u(x)+2u'(x){\big )}c'(x)=f(x)}$.

This reduces the order of the inhomogeneous differential equation. This becomes apparent when it is introduced so that applies ${\displaystyle z(x)=c'(x)}$

${\displaystyle u(x)z'(x)+{\big (}p(x)u(x)+2u'(x){\big )}z(x)=f(x)}$.

Division by supplies ${\displaystyle u(x)\neq 0}$

${\displaystyle z'(x)+{\Bigg (}p(x)+{\frac {2u'(x)}{u(x)}}{\Bigg )}z(x)={\frac {f(x)}{u(x)}}}$.

The further calculation requires the integrating factor

${\displaystyle \mu (x)=e^{\int _{a}^{x}({\frac {2u'(t)}{u(t)}}+p(t))\mathrm {d} t}=e^{\int _{a}^{x}(\log u^{2}(t)+p(t))\mathrm {d} t}=e^{\int _{a}^{x}({\frac {\mathrm {d} \log u^{2}(t)}{\mathrm {d} t}}+p(t))\mathrm {d} t}=e^{\int _{a}^{x}\mathrm {d} \log u^{2}(t)}e^{\int _{a}^{x}p(t))\mathrm {d} t}=u^{2}(x)e^{\int _{a}^{x}p(t)\mathrm {d} t}}$,

where represents a total differential and the lower integration limit is to be chosen appropriately. After multiplication by the integrating factor, the inhomogeneous differential equation takes on the following form ${\displaystyle \mathrm {d} \log u^{2}(t)}$ ${\displaystyle a}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(z(x)u^{2}(x)e^{\int _{a}^{x}p(t)\mathrm {d} t})=u(x)f(x)e^{\int _{a}^{x}p(t)\mathrm {d} t}}$.

After integrating this equation, we get a solution for . A further integration of yields the sought solution of the (inhomogeneous) differential equation, omitting the integration constants ${\displaystyle z(x)}$ ${\displaystyle c'(x)}$ ${\displaystyle c'(x)}$

${\displaystyle y(x)=c(x)u(x)}$.

example

The homogeneous differential equation with non-constant coefficients is considered

${\displaystyle v''(x)-2xv'(x)-2v(x)=0}$.

A solution to this homogeneous differential equation is . The approach of varying the constants now yields ${\displaystyle u(x)=e^{x^{2}}}$ ${\displaystyle y(x)=c(x)e^{x^{2}}}$

${\displaystyle {\big (}(2+4x^{2})e^{x^{2}}c(x)+2xe^{x^{2}}c'(x)+e^{x^{2}}c''(x){\big )}-2x{\big (}2xe^{x^{2}}c(x)+e^{x^{2}}c'(x){\big )}-2e^{x^{2}}c(x)=0}$

and after rearranging according to derivatives of ${\displaystyle c(x)}$

${\displaystyle e^{x^{2}}{\big (}c''(x)+2xc'(x){\big )}=0}$.

Since and is, the homogeneous differential equation can be transformed into ${\displaystyle e^{x^{2}}\neq 0}$ ${\displaystyle z(x)=c'(x)}$

${\displaystyle {\frac {z'(x)}{z(x)}}+2x=0}$

and thus

${\displaystyle {\frac {\mathrm {d} \log z(x)}{\mathrm {d} x}}=-2x}$

or

${\displaystyle z(x)=e^{-x^{2}}}$.

Hence the second solution to the homogeneous differential equation is given by , thus ${\displaystyle c(x)=\int _{0}^{x}z(t)\mathrm {d} t}$

${\displaystyle c(x)=\int _{0}^{x}e^{-t^{2}}\mathrm {d} t={\frac {\sqrt {\pi }}{2}}\operatorname {erf} (x)}$.

Here means the Gaussian error function . ${\displaystyle \operatorname {erf} (x)}$

literature

• Gerald Teschl : Ordinary Differential Equations and Dynamical Systems (=  Graduate Studies in Mathematics . Volume 140 ). American Mathematical Society, Providence 2012, ISBN 978-0-8218-8328-0 ( mat.univie.ac.at ).