The Lagrange identity , named after Joseph Louis Lagrange (1736–1813), is used in solving ordinary second-order differential equations , particularly in Sturm-Liouville problems .
definition
The Lagrange identity for the functions , from the differentiability class and the coefficient functions ,
and is given by the storm-Liouville operator
ϕ
{\ displaystyle \ phi}
ψ
{\ displaystyle \ psi}
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{\ displaystyle \ phi, \ psi \ in C ^ {2} ((a, b), \ mathbb {R})}
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{\ displaystyle w, q \ in C ^ {0} ((a, b), \ mathbb {R})}
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{\ displaystyle p \ in C ^ {1} ((a, b), \ mathbb {R})}
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{\ displaystyle p, q> 0}
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{\ displaystyle {\ mathcal {L}} = {\ frac {1} {w}} \ left (- {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \, p \, { \ frac {\ mathrm {d}} {\ mathrm {d} x}} + q \ right)}
for which applies:
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{\ displaystyle {\ begin {aligned} \ phi {\ mathcal {L}} \ psi - \ psi {\ mathcal {L}} \ phi & = {\ frac {-1} {w}} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ bigg (} p (\ phi \ psi '- \ psi \ phi') {\ bigg)} \\ & = {\ frac {-1} { w}} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ bigg (} pW (\ phi, \ psi) {\ bigg)} \ end {aligned}}}
where means the Wronsky determinant of the functions .
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{\ displaystyle W (\ phi, \ psi)}
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{\ displaystyle \ phi, \ psi}
Derivation
Let be a Sturm-Liouville differential operator, then:
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{\ displaystyle {\ mathcal {L}}}
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{\ displaystyle \ phi {\ mathcal {L}} \ psi = \ phi {\ frac {-1} {w}} {\ bigg (} {\ frac {\ mathrm {d}} {\ mathrm {d} x }} \ left (p {\ frac {\ mathrm {d} \ psi} {\ mathrm {d} x}} \ right) -q \ psi {\ bigg)},}
and
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{\ displaystyle \ psi {\ mathcal {L}} \ phi = \ psi {\ frac {-1} {w}} {\ bigg (} {\ frac {\ mathrm {d}} {\ mathrm {d} x }} \ left (p {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} x}} \ right) -q \ phi {\ bigg)}.}
Subtracting the two equations gives:
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{\ displaystyle \ phi {\ mathcal {L}} \ psi - \ psi {\ mathcal {L}} \ phi = \ phi {\ frac {-1} {w}} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p {\ frac {\ mathrm {d} \ psi} {\ mathrm {d} x}} \ right) - \ psi {\ frac {-1} {w} } {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} x}} \ right).}
Now, using the product rule for derivatives , the term is not taken into account, the following representations can be calculated and . In this way it can be seen that the second term is the same in both derivatives and disappears when the difference is formed, i.e.:
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{\ displaystyle {\ tfrac {-1} {w}}}
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{\ displaystyle \ textstyle \ phi {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p {\ frac {\ mathrm {d} \ psi} {\ mathrm {d} x} } \ right) = {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p \ phi {\ frac {\ mathrm {d} \ psi} {\ mathrm {d} x} } \ right) - \ left (p {\ frac {\ mathrm {d} \ psi} {\ mathrm {d} x}} \ right) {\ frac {\ mathrm {d} \ phi} {\ mathrm {d } x}}}
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{\ displaystyle \ textstyle \ psi {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} x} } \ right) = {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p \ psi {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} x} } \ right) - \ left (p {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} x}} \ right) {\ frac {\ mathrm {d} \ psi} {\ mathrm {d } x}}}
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{\ displaystyle {\ begin {aligned} \ phi {\ mathcal {L}} \ psi - \ psi {\ mathcal {L}} \ phi & = {\ frac {-1} {w}} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p \ phi {\ frac {\ mathrm {d} \ psi} {\ mathrm {d} x}} \ right) - {\ frac {- 1} {w}} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ left (p \ psi {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} x }} \ right) \\\\ & = {\ frac {-1} {w}} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ bigg (} p \ left ( \ phi {\ frac {\ mathrm {d} \ psi} {\ mathrm {d} x}} - \ psi {\ frac {\ mathrm {d} \ phi} {\ mathrm {d} x}} \ right) {\ bigg)} \\\\ & = {\ frac {-1} {w}} {\ frac {\ mathrm {d}} {\ mathrm {d} x}} {\ bigg (} pW (\ phi , \ psi) {\ bigg)}. \\\ end {aligned}}}
literature
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