Paradoxical Euler or Integration by Differentiation:
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1 Prdoicl Euler or Integrtion by Differentition: A Synopsis of E36 Andrew Fbin nd Hieu Nguyen Rown University The Euler Society 008 Annul Conference July 1, 008
2 E36 Eposition de quelques prdoes dns le clcul integrl (Eplntion of Certin Prdoes in Integrl Clculus) Originlly published in Memoires de l'cdemie des sciences de Berlin 1, 1758, pp ; Oper Omni: Series 1, Volume, pp (Avilble t The Euler Archive: ) English trnsltion by Andrew Fbin (007). Student Trnsltions of Euler s Mthemtics (STEM) Project - 7 students - 3 fculty members - 9 trnsltions
3 The First Prdo 3
4 4
5 PROBLEM I (E36) Given point A, find the curve EM such tht the perpendiculr AV, derived from point A onto some tngent of the curve MV, is the sme size everywhere. Mm, PMS, nd APR re similr y PS PM M Mm PM M PS Mm y ds PR AP m Mm AP m PR Mm dy ds AP Pp M ds Mm dy y PM dy m AV ( Tngentil distnce) 5
6 AV PS PR y ds dy ds y dy ds dy y ds Euler s Differentil Eqution: dy y dy Note: If the solution curve is ssumed to lie below -is, then Euler s DE becomes ( y) dy ( ) ( dy) dy y dy 6
7 Ordinry Method of Integrtion Euler s differentil eqution (differentil form): 1. Squre both sides: y dy dy y ydy dy dy. Solve for dy (Euler ignores negtive solution): 3. Rewrite: dy y y dy dy y y 7
8 4. u-substitution: LHS: dy dy y y y u ; dy du u u du ( ) u ( ) dy y ( ) RHS: Simplifies to: du 3 ( ) u u y ( )( u 1) du u 3 ( ) ( )( 1) ( ) du u 1 8
9 Cse I: Thus: u 1 ( ) du u 1 (true) y u y (Circle) Cse II: u 1 u du 1 log 1 log 1 u u n 9
10 u u 1 n n u( u u 1) 1 n un n 1 u n 1 1 n y 1 ( )( ) n n 1 ( ) y ( ) n ( n 1) ( n 1) y (Line) n n 10
11 Integrting by Differentiting Euler s Differentil Eqution: 1. Rewrite in terms of y dy dy p dy : y p 1 p. Differentite: dp Cse I: 0 dy p 1 p p p dp p 1 p 1 p dp p dp 1 p y 1 p p dp 1 p 11
12 Eliminte the prmeter: y Solution to Cse I: p 1 p 1 p p 1 y (Circle) p E A VM 1
13 dp Cse II: 0 Thus p = constnt = n y p 1 p y n 1 n (Tngent line) 13
14 Cubic: dp Cse I: 0 Higher-Order Emples y dy dy 3 3 y p 1 p p p 3 (1 p ) 3 y dy 3 (1 p ) 3 dp Cse II: 0 Thus p = constnt = n y p 3 1 p y n 1 n (Tngent line) 14
15 Arbitrry order: n n n n y dy ( dy dy etc.) n y p ( p p etc.) dp Cse I: 0 y dp Cse II: p p etc. n n ( p p etc.) n n ( n ) p ( n ) p etc. n 1 n ( p p etc.) n 1 Thus p = constnt = m n y m ( m m etc.) 15
16 Generl Solution of Euler s DE y p F p p Integrting by Differentiting: dp Cse I: 0 F'( p) dy dy dp 1 p F'( p) dp dp dp p p F '( p) F p pf '( p) y F p p (Prmetric solution!) dp Cse II: 0 y p F p Thus p = constnt (Tngent lines to solution in Cse I) 16
17 PROBLEM II (E36) On the is AB, find the curve AMB such tht hving derived from one of its points M the tngent TMV, it intersects the two lines AE nd BF, derived perpendiculrly to the is AB t the two given points A nd B, so tht the rectngle formed by the lines AT nd BV is the sme size everywhere. Mm, PMS, nd APR re similr AP y PM y Pp M dy m ds Mm dy AT BV TR m dy RM M MS m ( ) dy SV M c ( Tngentil re ) AB 17
18 AT PM RM y dy BV BS SV y ( ) dy Euler s Differentil Eqution AT BV y dy y dy dy y p p c p F( p) p c dy 18
19 Integrting by Differentiting Cse I: y dp 0 c c c p p p Eliminting the prmeter: ( ) p c p y c c c p ( ) y c c c p p 1 19
20 Solution to Cse I: ( ) y c 1 (Ellipse) 0
21 dp Cse II: 0 Thus p = constnt = n y p p y n AT n ( ) c n c c n p (Tngent line) BV AT n BV c c n 1
22 PROBLEM III (E36) Given two points A nd C, find the curve EM such tht if we derive some tngent MV, which the perpendiculr AV is directed towrds from the first point A, nd we join the stright line CV to V from the other point C, this line CV is the sme size everywhere. b =AP y=pm CV= AC=b
23 PROBLEM IV (E36) Given two points A nd B, find the curve EM such tht hving derived some tngent VMX, if the perpendiculrs AV nd BX re directed towrds it from the points A nd B, the rectngle of these lines is the sme size everywhere. =AP y=pm AB = b AV BX c 3
24 Generliztion to 3-D -D: (Distnce from curve to point), y dy, dy y dy dy dy dy y 1 3-D: (Distnce from surfce to point) zdy dydz ydz dy dz dy dz z z z z z y 1 y y 4
25 The Second Prdo 5
26 References Originlly published in Memoires de l'cdemie des sciences de Berlin 1, 1758, pp ; Oper Omni: Series 1, Volume, pp (Avilble t The Euler Archive: ) Student Trnsltions of Euler s Mthemtics (STEM) Project Website: 6
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