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August  2018, 11(4): 655-666. doi: 10.3934/dcdss.2018040

## Conditional symmetries of nonlinear third-order ordinary differential equations

 a. International Institute for Symmetry Analysis and Mathematical Modeling, North-West University, Mafikeng Campus, P Bag X2046, Mafikeng, South Africa b. School of Computer Science and Applied Mathematics DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa

* Corresponding author: Aeeman Fatima.

Received  December 2016 Revised  April 2017 Published  November 2017

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

Citation: Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040
##### References:
 [1] B. Abraham-Shrauner, K. S. Govinder and P. G. L. Leach, Integration of second order ordinary differential equations not possessing Lie point symmetries, Phys. Lett. A, 203 (1995), 169-174. doi: 10.1016/0375-9601(95)00426-4. Google Scholar [2] D. J. Arrigo and J. M. Hill, Nonclassical symmetries for nonlinear diffusion and absorption, Stud. Appl. Math., 94 (1995), 21-39. doi: 10.1002/sapm199594121. Google Scholar [3] G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025-1042. Google Scholar [4] S. S. Chern, Sur la géométrie d'une équation différentielle du troiséme ordre, CR Acad Sci Paris, 1937.Google Scholar [5] S. S. Chern, The geometry of the differential equation $y''''=F(x, y, y'', y''')$, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111. Google Scholar [6] R. Cherniha and M. Henkel, On non-linear partial differential equations with an infinite-dimensional conditional symmetry, J. Math. Anal. Appl., 298 (2004), 487-500. doi: 10.1016/j.jmaa.2004.05.038. Google Scholar [7] R. Cherniha and O. Pliukhin, New conditional symmetries and exact solutions of reaction-diffusion-convection equations with exponential nonlinearities, J. Math. Anal. Appl., 403 (2013), 23-37. doi: 10.1016/j.jmaa.2013.02.010. Google Scholar [8] P. A. Clarkson, Nonclassical symmetry reductions of the Boussinesq equation, Chaos Solitons Fractals, 5 (1995), 2261-2301. doi: 10.1016/0960-0779(94)E0099-B. Google Scholar [9] P. A. Clarkson, Nonclassical symmetry reductions of nonlinear partial differential equations, Math. Comput. Model., 18 (1993), 45-68. doi: 10.1016/0895-7177(93)90214-J. Google Scholar [10] P. L. Da Silva and I. L. Freire, Symmetry analysis of a class of autonomous even-order ordinary differential equations, IMA J. Appl. Math., 80 (2015), 1739-1758, arXiv: 1311.0313v2 [mathph] 7 march 2014. doi: 10.1093/imamat/hxv014. Google Scholar [11] A. Fatima and F. M. Mahomed, Conditional symmetries for ordinary differential equations and applications, Int. J. Non-Linear Mech., 67 (2014), 95-105. doi: 10.1016/j.ijnonlinmec.2014.08.013. Google Scholar [12] W. I. Fushchich, Conditional symmetry of equations of nonlinear mathematical physics, Ukrain. Math. Zh., 43 (1991), 1456-1470. doi: 10.1007/BF01067273. Google Scholar [13] G. Gaeta, Conditional symmetries and conditional constants of motion for dynamical systems, Report of the Centre de Physique Theorique Ecole Polytechnique, Palaiseau France, 1 (1993), 1-24. Google Scholar [14] A. Goriely, Integrability and Nonintegrability of Dynamical Systems, Advanced Series in Nonlinear Dynamics, 19. World Scientific Publishing Co. , Inc. , River Edge, NJ, 2001. doi: 10.1142/9789812811943. Google Scholar [15] G. Grebot, The characterization of third order ordinary differential equations admitting a transitive fibre-preserving point symmetry group, J. Math. Anal. Appl., 206 (1997), 364-388. doi: 10.1006/jmaa.1997.5219. Google Scholar [16] N. H. Ibragimov and S. V. Meleshko, Linearization of third order ordinary differential equations by point and contact transformations, J. Math. Anal. Appl., 308 (2005), 266-289. doi: 10.1016/j.jmaa.2005.01.025. Google Scholar [17] N. H. Ibragimov, S. V. Meleshko and S. Suksern, Linearization of fourth order ordinary differential equation by point transformations, J. Phys. A, 41 (2008), 235206, 19 pp. doi: 10.1088/1751-8113/41/23/235206. Google Scholar [18] A. H. Kara and F. M. Mahomed, A Basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys., 9 (2002), 60-72. doi: 10.2991/jnmp.2002.9.s2.6. Google Scholar [19] M. Kunzinger and R. O. Popovych, Generalized conditional symmetries of evolution equations, J. Math. Anal. Appl., 379 (2011), 444-460. doi: 10.1016/j.jmaa.2011.01.027. Google Scholar [20] P. G. L. Leach, Equivalence classes of second-order ordinary differential equations with three-dimensional Lie algebras of point symmetries and linearisation, J. Math. Anal. Appl., 284 (2003), 31-48. doi: 10.1016/S0022-247X(03)00147-1. Google Scholar [21] S. Lie, Lectures on Differential Equations with Known Infinitesimal Transformations, Leipzig, Teubner, 1981 (in German Lie's Lectures by G. Sheffers).Google Scholar [22] F. M. Mahomed, I. Naeem and A. Qadir, Conditional linearizability criteria for a system of third-order ordinary differential equations, Nonlinear Anal. B: Real World Appl., 10 (2009), 3404-3412. doi: 10.1016/j.nonrwa.2008.09.021. Google Scholar [23] F. M. Mahomed, Symmetry group classification of ordinary differential equations: Survey of some results, Math. Meth. Appl. Sci., 30 (2007), 1995-2012. doi: 10.1002/mma.934. Google Scholar [24] F. M. Mahomed and A. Qadir, Classification of ordinary differential equations by conditional linearizability and symmetry, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 573-584. doi: 10.1016/j.cnsns.2011.06.012. Google Scholar [25] F. M. Mahomed and A. Qadir, Conditional linearizability criteria for third order ordinary differential equations, J. Nonlinear Math. Phys., 15 (2008), 124-133. doi: 10.2991/jnmp.2008.15.s1.11. Google Scholar [26] F. M. Mahomed and A. Qadir, Conditional linearizability of fourth-order semilinear ordinary differential equations, J. Nonlinear Math. Phys., 16 (2009), 165-178. doi: 10.1142/S140292510900039X. Google Scholar [27] F. M. Mahomed and P. G. L. Leach, Symmetry Lie algebras of $n$ order ordinary differential equations, J. Math. Anal. Appl., 151 (1990), 80-107. doi: 10.1016/0022-247X(90)90244-A. Google Scholar [28] S. V. Meleshko, On linearization of third order ordinary differential equation, J. Phys. A, 39 (2006), 15135-15145. doi: 10.1088/0305-4470/39/49/005. Google Scholar [29] S. Neut and M. Petitot, La géométrie de l'équation $y'''=f(x,y,y',y'')$, CR Acad. Sci. Paris Sér. I., 335 (2002), 515-518. doi: 10.1016/S1631-073X(02)02507-4. Google Scholar [30] P. J. Olver and E. M. Vorob'ev, Nonclassical and conditional symmetries, in: N. H. Ibragiminov (Ed. ), CRC Handbook of Lie Group Analysis, vol. 3, CRC Press, Boca Raton, 1994.Google Scholar [31] E. Pucci and G. Saccomandi, Evolution equations, invariant surface conditions and functional separation of variables, Physica D: Nonlinear Phenomena, 139 (2000), 28-47. doi: 10.1016/S0167-2789(99)00224-9. Google Scholar [32] E. Pucci and G. Saccomandi, On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl., 163 (1992), 588-598. doi: 10.1016/0022-247X(92)90269-J. Google Scholar [33] E. Pucci, Similarity reductions of partial differential equations, J. Phys. A, 25 (1992), 2631-2640. doi: 10.1088/0305-4470/25/9/032. Google Scholar [34] W. Sarlet, P. G. L. Leach and F. Cantrijn, First integrals versus configurational invariants and a weak form of complete integrability, Physica D, 17 (1985), 87-98. doi: 10.1016/0167-2789(85)90136-8. Google Scholar [35] S. Spichak and V. Stognii, Conditional symmetry and exact solutions of the Kramers equation, Nonlinear Math. Phys., 2 (1997), 450-454. Google Scholar [36] S. Suksern, N. H. Ibragimov and S. V. Meleshko, Criteria for the fourth order ordinary differential equations to be linearizable by contact transformations, Common. Nonlinear Sci. Number. Simul., 14 (2009), 2619-2628. doi: 10.1016/j.cnsns.2008.09.021. Google Scholar [37] C. Wafo Soh and F. M. Mahomed, Linearization criteria for a system of second-order ordinary differential equations, Int. J. Non-Linear Mech., 36 (2001), 671-677. doi: 10.1016/S0020-7462(00)00032-9. Google Scholar

show all references

##### References:
 [1] B. Abraham-Shrauner, K. S. Govinder and P. G. L. Leach, Integration of second order ordinary differential equations not possessing Lie point symmetries, Phys. Lett. A, 203 (1995), 169-174. doi: 10.1016/0375-9601(95)00426-4. Google Scholar [2] D. J. Arrigo and J. M. Hill, Nonclassical symmetries for nonlinear diffusion and absorption, Stud. Appl. Math., 94 (1995), 21-39. doi: 10.1002/sapm199594121. Google Scholar [3] G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025-1042. Google Scholar [4] S. S. Chern, Sur la géométrie d'une équation différentielle du troiséme ordre, CR Acad Sci Paris, 1937.Google Scholar [5] S. S. Chern, The geometry of the differential equation $y''''=F(x, y, y'', y''')$, Sci. Rep. Nat. Tsing Hua Univ., 4 (1940), 97-111. Google Scholar [6] R. Cherniha and M. Henkel, On non-linear partial differential equations with an infinite-dimensional conditional symmetry, J. Math. Anal. Appl., 298 (2004), 487-500. doi: 10.1016/j.jmaa.2004.05.038. Google Scholar [7] R. Cherniha and O. Pliukhin, New conditional symmetries and exact solutions of reaction-diffusion-convection equations with exponential nonlinearities, J. Math. Anal. Appl., 403 (2013), 23-37. doi: 10.1016/j.jmaa.2013.02.010. Google Scholar [8] P. A. Clarkson, Nonclassical symmetry reductions of the Boussinesq equation, Chaos Solitons Fractals, 5 (1995), 2261-2301. doi: 10.1016/0960-0779(94)E0099-B. Google Scholar [9] P. A. Clarkson, Nonclassical symmetry reductions of nonlinear partial differential equations, Math. Comput. Model., 18 (1993), 45-68. doi: 10.1016/0895-7177(93)90214-J. Google Scholar [10] P. L. Da Silva and I. L. Freire, Symmetry analysis of a class of autonomous even-order ordinary differential equations, IMA J. Appl. Math., 80 (2015), 1739-1758, arXiv: 1311.0313v2 [mathph] 7 march 2014. doi: 10.1093/imamat/hxv014. Google Scholar [11] A. Fatima and F. M. Mahomed, Conditional symmetries for ordinary differential equations and applications, Int. J. Non-Linear Mech., 67 (2014), 95-105. doi: 10.1016/j.ijnonlinmec.2014.08.013. Google Scholar [12] W. I. Fushchich, Conditional symmetry of equations of nonlinear mathematical physics, Ukrain. Math. Zh., 43 (1991), 1456-1470. doi: 10.1007/BF01067273. Google Scholar [13] G. Gaeta, Conditional symmetries and conditional constants of motion for dynamical systems, Report of the Centre de Physique Theorique Ecole Polytechnique, Palaiseau France, 1 (1993), 1-24. Google Scholar [14] A. Goriely, Integrability and Nonintegrability of Dynamical Systems, Advanced Series in Nonlinear Dynamics, 19. World Scientific Publishing Co. , Inc. , River Edge, NJ, 2001. doi: 10.1142/9789812811943. Google Scholar [15] G. Grebot, The characterization of third order ordinary differential equations admitting a transitive fibre-preserving point symmetry group, J. Math. Anal. Appl., 206 (1997), 364-388. doi: 10.1006/jmaa.1997.5219. Google Scholar [16] N. H. Ibragimov and S. V. Meleshko, Linearization of third order ordinary differential equations by point and contact transformations, J. Math. Anal. Appl., 308 (2005), 266-289. doi: 10.1016/j.jmaa.2005.01.025. Google Scholar [17] N. H. Ibragimov, S. V. Meleshko and S. Suksern, Linearization of fourth order ordinary differential equation by point transformations, J. Phys. A, 41 (2008), 235206, 19 pp. doi: 10.1088/1751-8113/41/23/235206. Google Scholar [18] A. H. Kara and F. M. Mahomed, A Basis of conservation laws for partial differential equations, J. Nonlinear Math. Phys., 9 (2002), 60-72. doi: 10.2991/jnmp.2002.9.s2.6. Google Scholar [19] M. Kunzinger and R. O. Popovych, Generalized conditional symmetries of evolution equations, J. Math. Anal. Appl., 379 (2011), 444-460. doi: 10.1016/j.jmaa.2011.01.027. Google Scholar [20] P. G. L. Leach, Equivalence classes of second-order ordinary differential equations with three-dimensional Lie algebras of point symmetries and linearisation, J. Math. Anal. Appl., 284 (2003), 31-48. doi: 10.1016/S0022-247X(03)00147-1. Google Scholar [21] S. Lie, Lectures on Differential Equations with Known Infinitesimal Transformations, Leipzig, Teubner, 1981 (in German Lie's Lectures by G. Sheffers).Google Scholar [22] F. M. Mahomed, I. Naeem and A. Qadir, Conditional linearizability criteria for a system of third-order ordinary differential equations, Nonlinear Anal. B: Real World Appl., 10 (2009), 3404-3412. doi: 10.1016/j.nonrwa.2008.09.021. Google Scholar [23] F. M. Mahomed, Symmetry group classification of ordinary differential equations: Survey of some results, Math. Meth. Appl. Sci., 30 (2007), 1995-2012. doi: 10.1002/mma.934. Google Scholar [24] F. M. Mahomed and A. Qadir, Classification of ordinary differential equations by conditional linearizability and symmetry, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 573-584. doi: 10.1016/j.cnsns.2011.06.012. Google Scholar [25] F. M. Mahomed and A. Qadir, Conditional linearizability criteria for third order ordinary differential equations, J. Nonlinear Math. Phys., 15 (2008), 124-133. doi: 10.2991/jnmp.2008.15.s1.11. Google Scholar [26] F. M. Mahomed and A. Qadir, Conditional linearizability of fourth-order semilinear ordinary differential equations, J. Nonlinear Math. Phys., 16 (2009), 165-178. doi: 10.1142/S140292510900039X. Google Scholar [27] F. M. Mahomed and P. G. L. Leach, Symmetry Lie algebras of $n$ order ordinary differential equations, J. Math. Anal. Appl., 151 (1990), 80-107. doi: 10.1016/0022-247X(90)90244-A. Google Scholar [28] S. V. Meleshko, On linearization of third order ordinary differential equation, J. Phys. A, 39 (2006), 15135-15145. doi: 10.1088/0305-4470/39/49/005. Google Scholar [29] S. Neut and M. Petitot, La géométrie de l'équation $y'''=f(x,y,y',y'')$, CR Acad. Sci. Paris Sér. I., 335 (2002), 515-518. doi: 10.1016/S1631-073X(02)02507-4. Google Scholar [30] P. J. Olver and E. M. Vorob'ev, Nonclassical and conditional symmetries, in: N. H. Ibragiminov (Ed. ), CRC Handbook of Lie Group Analysis, vol. 3, CRC Press, Boca Raton, 1994.Google Scholar [31] E. Pucci and G. Saccomandi, Evolution equations, invariant surface conditions and functional separation of variables, Physica D: Nonlinear Phenomena, 139 (2000), 28-47. doi: 10.1016/S0167-2789(99)00224-9. Google Scholar [32] E. Pucci and G. Saccomandi, On the weak symmetry groups of partial differential equations, J. Math. Anal. Appl., 163 (1992), 588-598. doi: 10.1016/0022-247X(92)90269-J. Google Scholar [33] E. Pucci, Similarity reductions of partial differential equations, J. Phys. A, 25 (1992), 2631-2640. doi: 10.1088/0305-4470/25/9/032. Google Scholar [34] W. Sarlet, P. G. L. Leach and F. Cantrijn, First integrals versus configurational invariants and a weak form of complete integrability, Physica D, 17 (1985), 87-98. doi: 10.1016/0167-2789(85)90136-8. Google Scholar [35] S. Spichak and V. Stognii, Conditional symmetry and exact solutions of the Kramers equation, Nonlinear Math. Phys., 2 (1997), 450-454. Google Scholar [36] S. Suksern, N. H. Ibragimov and S. V. Meleshko, Criteria for the fourth order ordinary differential equations to be linearizable by contact transformations, Common. Nonlinear Sci. Number. Simul., 14 (2009), 2619-2628. doi: 10.1016/j.cnsns.2008.09.021. Google Scholar [37] C. Wafo Soh and F. M. Mahomed, Linearization criteria for a system of second-order ordinary differential equations, Int. J. Non-Linear Mech., 36 (2001), 671-677. doi: 10.1016/S0020-7462(00)00032-9. Google Scholar
Lie group classification of scalar second-order equations in the real plane
 $p=\partial /\partial x$ and $q=\partial /\partial y$ Algebra Canonical forms of generators Representative equations $L_{1}$ $X_1=p$ $y''=g(y,y')$ $L_{2;1}^I$ $X_1=p,X_2=q$ $y''=g(y')$ $L_{2;1}^{II}$ $X_1=q,X_2=xp+yq$ $xy''=g(y')$ $L_{3;3}^I$ $X_1=p, X_2=q, X_3=xp+(x+y)q$ $y''=Ae^{-y'}$ $L_{3;6}^I$ $X_1=p, X_2=q, X_3=xp+ayq$ $y''=Ay'^{(a-2)/(a-1)}$ $L_{3;7}^I$ $X_1=p, X_2=q, X_3=(bx+y)p+(by-x)q$ $y''=A(1+y'^{2})^{\frac{3}{2}}e^{b\arctan y'}$ $L_{3;8}^I$ $X_1=q, X_2=xp+yq, X_3=2xyp+y^2q$ $xy''=Ay'^{3}-\frac{1}{2}y'$ $L_{3;8}^{II}$ $X_1=q, X_2=xp+yq, X_3=2xyp+(y^2+x^2)q$ $xy''=y'+y'^{3}+A(1+y'^{2})^{\frac{3}{2}}$ $L_{3;8}^{III}$ $X_1=q, X_2=xp+yq, X_3=2xyp+(y^2-x^2)q$ $xy''=y'-y'^{3}+A(1-y'^{2})^{\frac{3}{2}}$ $L_{3;9}$ $X_1=(1+x^2)p+xyq, X_2=xyp+(1+y^2)q$, $X_3=yp-xq$ $y''=A[\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}]^{3/2}$
 $p=\partial /\partial x$ and $q=\partial /\partial y$ Algebra Canonical forms of generators Representative equations $L_{1}$ $X_1=p$ $y''=g(y,y')$ $L_{2;1}^I$ $X_1=p,X_2=q$ $y''=g(y')$ $L_{2;1}^{II}$ $X_1=q,X_2=xp+yq$ $xy''=g(y')$ $L_{3;3}^I$ $X_1=p, X_2=q, X_3=xp+(x+y)q$ $y''=Ae^{-y'}$ $L_{3;6}^I$ $X_1=p, X_2=q, X_3=xp+ayq$ $y''=Ay'^{(a-2)/(a-1)}$ $L_{3;7}^I$ $X_1=p, X_2=q, X_3=(bx+y)p+(by-x)q$ $y''=A(1+y'^{2})^{\frac{3}{2}}e^{b\arctan y'}$ $L_{3;8}^I$ $X_1=q, X_2=xp+yq, X_3=2xyp+y^2q$ $xy''=Ay'^{3}-\frac{1}{2}y'$ $L_{3;8}^{II}$ $X_1=q, X_2=xp+yq, X_3=2xyp+(y^2+x^2)q$ $xy''=y'+y'^{3}+A(1+y'^{2})^{\frac{3}{2}}$ $L_{3;8}^{III}$ $X_1=q, X_2=xp+yq, X_3=2xyp+(y^2-x^2)q$ $xy''=y'-y'^{3}+A(1-y'^{2})^{\frac{3}{2}}$ $L_{3;9}$ $X_1=(1+x^2)p+xyq, X_2=xyp+(1+y^2)q$, $X_3=yp-xq$ $y''=A[\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}]^{3/2}$
Inherited symmetries of derived scalar third-order equations
 Representative 2nd-order ODE $A\ne0$ Derived 3rd-order ODE Inherited algebra $y''=Ae^{-y'}$ $y'''+y''^{2}=0$ $L_{3;3}^{I}$ $y''=Ay'^{(a-2)/(a-1)}$ $y'y'''-\frac{a-2}{a-1}y''^2=0$ $L_{3;6}^I$ $y''=A(1+y'^{2})^{\frac{3}{2}}e^{b\arctan y'}$ $y'''-\frac{3y'+b}{1+y'^2}y''^2=0$ $L_{3;7}^I$ $xy''=Ay'^{3}-\frac{1}{2}y'$ $y'y'''-3y''^2=0$ $L_{3;8}^I$ $xy''=y'+y'^{3}+A(1+y'^{2})^{\frac{3}{2}}$ $y'''+y'''y'^2-3y'y''^2=0$ $L_{3;8}^{II}$ $xy''=y'-y'^{3}+A(1-y'^{2})^{\frac{3}{2}}$ $y'''-y'''y'^2-3y'y''^2=0$ $L_{3;8}^{III}$ $y''=AK^{3/2}$ $y'''=\frac32y''K^{-1}D_x K$ $L_{3;9}$ where $K=\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}$
 Representative 2nd-order ODE $A\ne0$ Derived 3rd-order ODE Inherited algebra $y''=Ae^{-y'}$ $y'''+y''^{2}=0$ $L_{3;3}^{I}$ $y''=Ay'^{(a-2)/(a-1)}$ $y'y'''-\frac{a-2}{a-1}y''^2=0$ $L_{3;6}^I$ $y''=A(1+y'^{2})^{\frac{3}{2}}e^{b\arctan y'}$ $y'''-\frac{3y'+b}{1+y'^2}y''^2=0$ $L_{3;7}^I$ $xy''=Ay'^{3}-\frac{1}{2}y'$ $y'y'''-3y''^2=0$ $L_{3;8}^I$ $xy''=y'+y'^{3}+A(1+y'^{2})^{\frac{3}{2}}$ $y'''+y'''y'^2-3y'y''^2=0$ $L_{3;8}^{II}$ $xy''=y'-y'^{3}+A(1-y'^{2})^{\frac{3}{2}}$ $y'''-y'''y'^2-3y'y''^2=0$ $L_{3;8}^{III}$ $y''=AK^{3/2}$ $y'''=\frac32y''K^{-1}D_x K$ $L_{3;9}$ where $K=\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}$
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