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 .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![(n-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e)
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 .
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle L (y) = f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73431c55e094a85da4988b45105f70dad6477e35)
![{\ displaystyle L (u) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b235374e9925be3f1ce539e40b6ea5cf4e473d37)
![y (x): = c (x) u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c44aaba4ee1b659a8d2241d39e51bdef5cba4d)
![{\ displaystyle L (y) = f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73431c55e094a85da4988b45105f70dad6477e35)
![{\ displaystyle {\ tilde {L}} (c ') = f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ede44047123299315f98afb6b4050201ae996361)
![n-1](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521)
![c '(x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/68b09c201cc83578d96e087d3b6623933b102134)
Formulation of the sentence
Consider the -th order differential operator
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle L (v) (x): = \ sum _ {k = 0} ^ {n} a_ {k} (x) v ^ {(k)} (x) \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb2fc4b31914f503803e8bf213911f66d6c5cec)
For this, let us solve the homogeneous linear differential equation
![u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e5ff65a28eed29d36ddae9c6ae3b596fd14370)
![{\ displaystyle L (u) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b235374e9925be3f1ce539e40b6ea5cf4e473d37)
known. For
![y (x): = c (x) u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c44aaba4ee1b659a8d2241d39e51bdef5cba4d)
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 ^ {(kj-1)} (x) \ right] c ^ {(j + 1)} (x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/737c737450ba9e86e5e7f9a19e309786aad22478)
In other words: solves the inhomogeneous differential equation -th order if and only if
![y (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e871993bfd131a8b0c3591c26084cf8171a74dcd)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
![{\ displaystyle {\ mathcal {L}} (y) = f (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88ba9d91e930aefce27b877894b8cb2b978ca9df)
![z (x): = c '(x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a19daefebc948594268fe74b60fee89eb0614391)
the inhomogeneous linear differential equation -th order
![(n-1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e)
![\ sum _ {{j = 0}} ^ {{n-1}} \ left [\ sum _ {{k = j + 1}} ^ {n} {k \ choose {j + 1}} a_ {k } (x) u ^ {{(kj-1)}} (x) \ right] z ^ {{(j)}} (x) = f (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b89358580c838ad16153ea00deaa8b73141c0b8)
solves.
proof
According to Leibniz's rule, the following applies
![(c \ cdot u) ^ {{(k)}} (x) = \ sum _ {{j = 0}} ^ {k} {k \ choose j} c ^ {{(j)}} (x) u ^ {{(kj)}} (x) \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/b219f045eb4c6ebabd6be9dd2598d58926cd1ea7)
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 ^ {(kj)} (x) = \ sum _ {j = 0} ^ {n} \ sum _ {k = j} ^ {n} {\ binom {k} {j}} a_ {k} (x) u ^ {(kj)} (x ) c ^ {(j)} (x) \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/865156f775b43a15b6e6a10562bd45bdeb357399)
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 ^ {( kj)} (x) c ^ {(j)} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7fa51824f9bb597068a7127eaad02300305776)
![{\ displaystyle c ^ {(j)} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40d5d0a2ecef8972024293f6452bfb2210457a9f)
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0b4a5410cd5bbfc6ff5a9a9ebc56ed4e40bb15)
![j](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
![{\ 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 ^ {(kj)} (x) \ right] c ^ {(j)} (x) \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf84b6d2148f4fd30fe552870e6b42c5af7d379)
The index shift provides the result
-
,
or using
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.
![\Box](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
example
The homogeneous linear differential equation of the 2nd order with constant coefficients is given
-
.
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/939432577738dbc0d0f2e4df68a95dcd1ad87ffd)
![{\ displaystyle \ lambda _ {1,2} = - 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a3148d42bd7f4d2db47b90ba935bffc125fe726)
![{\ displaystyle u (x) = e ^ {- 2x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45708becc923991a42c6159eeaaea89980b78b48)
![{\ displaystyle y (x) = c (x) u (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4025292fe54cda24f0dc615200c0df93ec40403)
and the given differential equation is represented as follows
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By rearranging the differential equation according to the derivatives of, we get
![{\ displaystyle c (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b20e1deb5bbfe90c658811e9635d865f02902791)
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.
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79cb8ad59261d371630a40b1559a2a64950b094b)
![{\ displaystyle u (x) c '' (x) + \ left (2u '(x) + 4u (x) \ right) c' (x) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16cd7b091a3e5d6c555e4950af72e837038e5a25)
and results with the already known solution for the second term , so that the differential equation is reduced to
![{\ displaystyle u (x) = e ^ {- 2x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45708becc923991a42c6159eeaaea89980b78b48)
![{\ displaystyle 2u '(x) + 4u (x) = - 4e ^ {- 2x} + 4e ^ {- 2x} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52c3873647f8aa81ae3ee6473fcddb9fb26f4cf0)
-
.
Since the exponential function represents and is therefore greater than zero everywhere, the condition for the second solution is the differential equation
![u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e5ff65a28eed29d36ddae9c6ae3b596fd14370)
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By integrating twice, we get with the integration constants
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The approach for the second solution of the differential equation thus results
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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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ede7c76f477757fc9a69c4ef15bf69f985ff9b9)
![{\ displaystyle y (x) = xu (x) = xe ^ {- 2x}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1063f562a2336fc49663cc86822c16ce1ea11a5)
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86aea615f963a095726b86170f839a0a3bc534ee)
Special case: linear differential equation of the second order
Let the homogeneous linear differential equation of the second order be solved
![u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e5ff65a28eed29d36ddae9c6ae3b596fd14370)
![u '' (x) + p (x) u '(x) + q (x) u (x) = 0 \.](https://wikimedia.org/api/rest_v1/media/math/render/svg/99aa35e0df23572a46e81c8126bbe782ddc5cb4b)
Then
![y (x): = c (x) u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c44aaba4ee1b659a8d2241d39e51bdef5cba4d)
Solution of the (inhomogeneous) differential equation
![y '' (x) + p (x) y '(x) + q (x) y (x) = f (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/26da1b2bcf06bced30c6db684d6e15363bf9f7c5)
exactly when
![z (x): = c '(x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a19daefebc948594268fe74b60fee89eb0614391)
the equation
![{\ displaystyle u (x) z '(x) + {\ big (} p (x) u (x) + 2u' (x) {\ big)} z (x) = f (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23251727511a386909075bee0a7ff9c61844dd8d)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26da1b2bcf06bced30c6db684d6e15363bf9f7c5)
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
![u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e5ff65a28eed29d36ddae9c6ae3b596fd14370)
-
,
where is any function. So is
![c (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b20e1deb5bbfe90c658811e9635d865f02902791)
![{\ displaystyle y '(x) = c' (x) u (x) + c (x) u '(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b5f84064e86d1e15a256d922dca8f5c5379f8c)
and
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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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dba9831da420f655af61916571f072d5d6c446a)
and by rearranging according to the derivatives of
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Since there is a solution to the homogeneous differential equation , the inhomogeneous differential equation can be reduced by this term and it applies
![u (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e5ff65a28eed29d36ddae9c6ae3b596fd14370)
![{\ displaystyle u '' (x) + p (x) u '(x) + q (x) u (x) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9da05ad9f818d1ac680cc4ff3b82e935938b4a)
-
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This reduces the order of the inhomogeneous differential equation. This becomes apparent when it is introduced so that applies
![{\ displaystyle z (x) = c '(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24401fbeb8da2a920912cbc594a4bf91eef6e3c8)
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Division by supplies
![{\ displaystyle u (x) \ neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f43bcff828495f23648cb5425b9ea1810df9f8)
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The further calculation requires the integrating factor
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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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae433c43be111082e40495f44a86a279ca004de3)
![a](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
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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
![z (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0345a757205f8f9fb1ea08964d54113563760b)
![c '(x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/68b09c201cc83578d96e087d3b6623933b102134)
![c '(x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/68b09c201cc83578d96e087d3b6623933b102134)
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example
The homogeneous differential equation with non-constant coefficients is considered
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A solution to this homogeneous differential equation is . The approach of varying the constants now yields
![{\ displaystyle u (x) = e ^ {x ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dba97cb6094f1bad20609a0ce47a98c2ef9a092)
![{\ displaystyle y (x) = c (x) e ^ {x ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5806213320d49aa9ab728de6dabfcebeb566fdfe)
![{\ 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/303866cd02192b97deac93445db97490fe137c9b)
and after rearranging according to derivatives of
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Since and is, the homogeneous differential equation can be transformed into
![{\ displaystyle e ^ {x ^ {2}} \ neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d428c26c643c1ef7fedd9be79a964311f59bd6)
![{\ displaystyle z (x) = c '(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24401fbeb8da2a920912cbc594a4bf91eef6e3c8)
![{\ displaystyle {\ frac {z '(x)} {z (x)}} + 2x = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f82228274426c4412c48afe5de60b043fee9aa9)
and thus
![{\ displaystyle {\ frac {\ mathrm {d} \ log z (x)} {\ mathrm {d} x}} = - 2x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af43106ac724276402cf4ac1aa09789ee281ee93)
or
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Hence the second solution to the homogeneous differential equation is given by , thus
![{\ displaystyle c (x) = \ int _ {0} ^ {x} z (t) \ mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f13ba6b8f219f3933e5eff1e4a025e0845e74127)
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Here means the Gaussian error function .
![{\ displaystyle \ operatorname {erf} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a96a1b139214ea50c6d6f436fb555e6429134e)
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