The Euler equation (by Leonhard Euler ) is a linear differential equation of higher order with non-constant coefficients of the particular form
![{\ displaystyle \ sum _ {k = 0} ^ {N} a_ {k} \, (cx + d) ^ {k} \; y ^ {(k)} (x) = b (x) \, \ cx + d> 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/338cae46e8b791c263d2a1fc8ed496cee056ae3f)
to given and inhomogeneity . If one knows a fundamental system of the homogeneous solution, one can determine the general solution of the inhomogeneous equation with the procedure of the variation of the constants . Therefore only needs to be considered.
![{\ displaystyle N \ in \ mathbb {N}, \ a_ {0}, \ ldots, a_ {N}, c, d \ in \ mathbb {R}, \ c \ neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a7c9fb7ba9065b54269fb1e113683f36356e6e0)
![b](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
![{\ displaystyle b \ equiv 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e51fb8fe1639251db987f5833dba289e5f8ffd5)
Euler's differential equation is converted into a linear differential equation with constant coefficients by means of the transformation .
![{\ displaystyle z (t): = y \ left ({\ tfrac {e ^ {t} -d} {c}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3299a143304b62214a93dae1f618dc8f52bcfac6)
Motivation of the transformation
Be a sufficiently smooth function and
![y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d)
-
, so .![{\ displaystyle \ y (x) = z (\ ln (cx + d))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cff531d5bb1854283cb025a38d12f0e50a82320d)
Then applies
![{\ displaystyle {\ begin {array} {lll} y '(x) & = & {\ frac {c} {cx + d}} z' (\ ln (cx + d)) \, \\ y '' (x) & = & {\ frac {c ^ {2}} {(cx + d) ^ {2}}} z '' (\ ln (cx + d)) - {\ frac {c ^ {2} } {(cx + d) ^ {2}}} z '(\ ln (cx + d)) \, \\\ end {array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12d266dcb592b7d941d843f1eee458814b345841)
so
![{\ displaystyle {\ begin {array} {lll} (cx + d) y '(x) & = & c \ cdot z' (\ ln (cx + d)) \, \\ (cx + d) ^ {2 } y '' (x) & = & c ^ {2} \ cdot [z '' - z '] (\ ln (cx + d)) \. \\\ end {array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23e119e239011b2711ee118e4c54e32bb279b7e7)
In this respect, Euler's second order differential equation would transform into a linear differential equation with constant coefficients. The following questions now arise:
- Does this transformation also convert the higher order terms into those with constant coefficients?
![{\ displaystyle (cx + d) ^ {k} y ^ {(k)} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e76c0e9ca47e6fdb998e4b0818587c4e658a9073)
- How can one calculate the coefficients on the right-hand side more easily without deriving the transformation enough times each time?
These questions are clarified by the following transformation theorem:
The transformation set
Let the linear differential equation be solved with constant coefficients
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle \ sum _ {k = 0} ^ {n} a_ {k} c ^ {k} \ left (\ left [\ prod _ {j = 0} ^ {k-1} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right] z \ right) (x) = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/511fab8454384a7bce2c6209d67bb74dd45a4a0a)
Then
![{\ displaystyle \ y (x): = z (\ ln (cx + d))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e381e2a0234eb0c36aa78eb16caa40b0d92a030)
a solution of Euler's (homogeneous) differential equation
![{\ displaystyle \ sum _ {k = 0} ^ {N} a_ {k} (cx + d) ^ {k} y ^ {(k)} (x) = 0 \, \ cx + d> 0 \. }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a09fdb9dbefd35e57f217fa5f9599e6ece7c13bf)
Explanation of the notation
Here, the differential operators are first linked with each other (comparable to multiplying) before they are applied to a function, for example:
![{\ displaystyle \ left [\ prod _ {j = 0} ^ {- 1} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right ] z = z \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d619c881c11df8c63e9b8243f3a187c8882aefe3)
![{\ displaystyle \ left [\ prod _ {j = 0} ^ {0} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right] z = \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - 0 \ right) z = z '\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0cd43709defaf2512b1b703d30b03db6b1730c)
![{\ displaystyle \ left [\ prod _ {j = 0} ^ {1} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right] z = \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - 0 \ right) \ left ({\ frac {\ rm {d}} {{\ rm {d }} x}} - 1 \ right) z = \ left ({\ frac {\ rm {d ^ {2}}} {{\ rm {d}} x ^ {2}}} - {\ frac {\ rm {d}} {{\ rm {d}} x}} \ right) z = z '' - z '\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/034228fbe23f1dcf8a2cb63d2f2fb34f2d538359)
![{\ displaystyle \ left [\ prod _ {j = 0} ^ {2} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right] z = \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - 0 \ right) \ left ({\ frac {\ rm {d}} {{\ rm {d }} x}} - 1 \ right) \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - 2 \ right) z = \ left ({\ frac {\ rm {d ^ {3}}} {{\ rm {d}} x ^ {3}}} - 3 {\ frac {\ rm {d ^ {2}}} {{\ rm {d}} x ^ { 2}}} + 2 {\ frac {\ rm {d}} {{\ rm {d}} x}} \ right) z = z '' '- 3z' '+ 2z' \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/069495c4ab6c049d06ba964ba0eaa7a50304936a)
proof
To show is only for everyone . This is done by means of complete induction . The beginning of induction is trivial. Provided that the identity is valid for , this identity can be differentiated. It turns out
![{\ displaystyle c ^ {k} \ left (\ left [\ prod _ {j = 0} ^ {k-1} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x }} - j \ right) \ right] z \ right) (\ ln (cx + d)) = (cx + d) ^ {k} y ^ {(k)} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/200b5517f2019285ed4c048f1ecf903645a5b7f7)
![k \ in \ mathbb {N} _0](https://wikimedia.org/api/rest_v1/media/math/render/svg/97bceb13f72e37bcd50b60e5fb2fa05bcf15c265)
![k = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b)
![{\ displaystyle k_ {0} \ in \ mathbb {N} _ {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0bfd65773d8f9be03e63918f62c2a97645875c9)
![{\ displaystyle (cx + d) ^ {k_ {0}} y ^ {(k_ {0} +1)} (x) + ck_ {0} (cx + d) ^ {k_ {0} -1} y ^ {(k_ {0})} (x) = {\ frac {c ^ {k_ {0} +1}} {cx + d}} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} \ left [\ prod _ {j = 0} ^ {k_ {0} -1} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x }} - j \ right) \ right] z \ right) (\ ln (cx + d)) \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba355907b3a201ec26d2fb5ab1447eae9dd634b1)
Applying the induction hypothesis implies
![{\ displaystyle {\ begin {array} {lll} (cx + d) ^ {k_ {0} +1} y ^ {(k_ {0} +1)} (x) & = & c ^ {k_ {0} +1} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} \ left [\ prod _ {j = 0} ^ {k_ {0} -1} \ left ( {\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right] z \ right) (\ ln (cx + d)) - ck_ {0} (cx + d) ^ {k_ {0}} y ^ {(k_ {0})} (x) \\ & = & c ^ {k_ {0} +1} \ left ({\ frac {\ rm {d}} { {\ rm {d}} x}} \ left [\ prod _ {j = 0} ^ {k_ {0} -1} \ left ({\ frac {\ rm {d}} {{\ rm {d} } x}} - j \ right) \ right] z \ right) (\ ln (cx + d)) \\ && \ quad -c ^ {k_ {0} +1} k_ {0} \ left (\ left [\ prod _ {j = 0} ^ {k_ {0} -1} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right] z \ right) (\ ln (cx + d)) \\ & = & c ^ {k_ {0} +1} \ left (\ left [\ prod _ {j = 0} ^ {k_ {0}} \ left ({\ frac {\ rm {d}} {{\ rm {d}} x}} - j \ right) \ right] z \ right) (\ ln (cx + d)) \. \\\ end { array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc1981dd15fb49a36f99a96d34cd701b97d4d2ad)
![\Box](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Conclusion: construction of a fundamental system
The characteristic equation for the differential equation of is
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle \ chi (\ lambda) = \ sum _ {k = 0} ^ {n} a_ {k} c ^ {k} \ prod _ {j = 0} ^ {k-1} (\ lambda -j ) = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34465b97146586aa8ee61c56ed35d9e6ed98a24d)
Now denote the zeros of the characteristic polynomial and the multiplicity of , so forms
![{\ displaystyle \ lambda _ {1}, \ ldots, \ lambda _ {M}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb60ec3ea1827fb57f82d61e48b22a3659c8d67f)
![{\ displaystyle \ chi (\ lambda)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34d21fb97a921d9c847ed2769c0d0f702b3694e6)
![R_j](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc93d61e1436bb2c2fb771a13d0892784754998)
![\ lambda _ {j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa91daf9145f27bb95746fd2a37537342d587b77)
![{\ displaystyle \ {z_ {j, k} (x) = e ^ {\ lambda _ {j} z} z ^ {k} \ | \ j = 1, \ ldots, M \, \ k = 0, \ ldots, R_ {j} -1 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9db7e27b44c52da9d5e6e756acdc5f7b74f6e24a)
a fundamental system of the equation for . So is
![z](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98)
![{\ displaystyle \ {y_ {j, k} (x) = (cx + d) ^ {\ lambda _ {j}} [\ ln (cx + d)] ^ {k} \ | \ j = 1, \ ldots, M \, \ k = 0, \ ldots, R_ {j} -1 \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/822649e4c3c847d87112aa740bd4f0a6c47b29fc)
a fundamental system of the (homogeneous) Euler differential equation.
example
Euler's differential equation is given
![{\ displaystyle a_ {2} x ^ {2} y '' (x) + a_ {1} xy '(x) + a_ {0} y (x) = 0 \, \ a_ {2} \ neq 0 \ , \ x> 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/751dae590ff6398f150a936da50da2ebc3fc0e0d)
According to the above sentence, the following linear differential equation with constant coefficients has to be solved
![{\ displaystyle a_ {2} (z '' (x) -z '(x)) + a_ {1} z' (x) + a_ {0} z (x) = 0 \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e572454c203bf3bbe419498ecdb8473d08752916)
so
![{\ displaystyle a_ {2} z '' (x) + (a_ {1} -a_ {2}) z '(x) + a_ {0} z (x) = 0 \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/222882b5edf4340b44db5bf66740deded3d2233c)
The characteristic polynomial belonging to this differential equation is
![{\ displaystyle \ chi (\ lambda) = \ a_ {2} \ lambda ^ {2} + (a_ {1} -a_ {2}) \ lambda + a_ {0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34ead342690bde6ea2d4a442fa855886ab7ea494)
and has the zeros
![{\ displaystyle \ lambda _ {1,2} = {\ frac {a_ {2} -a_ {1}} {2a_ {2}}} \ pm {\ sqrt {{\ frac {(a_ {2} -a_ {1}) ^ {2}} {4a_ {2} ^ {2}}} - {\ frac {a_ {0}} {a_ {2}}}}} \.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae8dc6472b34dc355483c7f1e3f5334a8487cf8)
Case 1 :, both real.
![{\ displaystyle \ lambda _ {1} \ neq \ lambda _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efc7088b0b1ade20fc3b36d596d0a11faaafca83)
Then there is a fundamental system for the transformed linear differential equation. The inverse transformation yields that there is
a fundamental system for the original Euler's differential equation.
![{\ displaystyle \ {e ^ {\ lambda _ {1} z}, e ^ {\ lambda _ {2} z} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0db6a36c97dc13cb96e3316893947901f3e7b23)
![{\ displaystyle \ {x ^ {\ lambda _ {1}}, x ^ {\ lambda _ {2}} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/136465a1f633829ef25f653f980cafa2ebf35eac)
Case 2: .
![{\ displaystyle \ \ lambda _ {1} = \ lambda _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bf39517a39e785f6156eb713c8fce677a549a99)
Then there is a double zero of the characteristic polynomial. Hence is a fundamental system for the transformed linear differential equation. The inverse transformation yields that there is
a fundamental system for the original Euler's differential equation.
![{\ displaystyle \ lambda: = {\ frac {a_ {2} -a_ {1}} {2a_ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a24e29d3e91d2190f79457faabbeacc2b5cf8f2)
![{\ displaystyle \ \ {e ^ {\ lambda z}, ze ^ {\ lambda z} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b84515013c621ad22e6d360f2663a505f975718d)
![{\ displaystyle \ \ {x ^ {\ lambda}, x ^ {\ lambda} \ ln x \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c368563776bcf47cde897395981cba0990c8699)
Case 3: both not real.
![{\ displaystyle \ \ lambda _ {1}, \ lambda _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34eed7ff8a1d1d5cf55783eea6b35407960cc6e7)
Then are complex conjugate to each other. So is a (complex) fundamental system. Be , . Then is a real fundamental system of the transformed linear differential equation. Inverse transformation provides the fundamental system for the original Euler's differential equation.
![{\ displaystyle \ \ lambda _ {1}, \ lambda _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34eed7ff8a1d1d5cf55783eea6b35407960cc6e7)
![{\ displaystyle \ \ {e ^ {\ lambda _ {1} z}, e ^ {\ lambda _ {2} z} \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79f47535264e06d6765bd8f8fb60c591c5858554)
![{\ displaystyle \ \ lambda _ {1} = \ mu + i \ nu}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b03bee5da5f62e2e12f32698821d1e33bb69068)
![{\ displaystyle \ mu, \ nu \ in \ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c9c1c9a25ab31642af416e9fc09c09d47ac54ea)
![{\ displaystyle \ \ {e ^ {\ mu z} \ sin (\ nu z), e ^ {\ mu z} \ cos (\ nu z) \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6200a03dba79c22c14674c0f6e5195eb2bb7232a)
![{\ displaystyle \ \ {x ^ {\ mu} \ sin (\ nu \ ln x), x ^ {\ mu} \ cos (\ nu \ ln x) \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a71e30e001be414e5a50b0fd716f4ee5fa1e276)
![\Box](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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