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Conditional symmetries of nonlinear third-order ordinary differential equations

  • * Corresponding author: Aeeman Fatima.

    * Corresponding author: Aeeman Fatima. 
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  • 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.

    Mathematics Subject Classification: 34A05, 22E05, 22E45, 22E60, 22E70.


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  • Table Ⅰ.  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
    $L_{1}$ $X_1=p$ $y''=g(y,y')$
    $L_{2;1}^I$ $X_1=p,X_2=q$ $y''=g(y')$
    $L_{2;1}^{II}$ $X_1=q,X_2=xp+yq$ $xy''=g(y')$
    $L_{3;3}^I$ $X_1=p, X_2=q, X_3=xp+(x+y)q$ $y''=Ae^{-y'}$
    $L_{3;6}^I$ $X_1=p, X_2=q, X_3=xp+ayq$ $y''=Ay'^{(a-2)/(a-1)}$
    $L_{3;7}^I$ $X_1=p, X_2=q, X_3=(bx+y)p+(by-x)q$ $y''=A(1+y'^{2})^{\frac{3}{2}}e^{b\arctan y'}$
    $L_{3;8}^I$ $X_1=q, X_2=xp+yq, X_3=2xyp+y^2q$ $xy''=Ay'^{3}-\frac{1}{2}y'$
    $L_{3;8}^{II}$ $X_1=q, X_2=xp+yq, X_3=2xyp+(y^2+x^2)q$ $xy''=y'+y'^{3}+A(1+y'^{2})^{\frac{3}{2}}$
    $L_{3;8}^{III}$ $X_1=q, X_2=xp+yq, X_3=2xyp+(y^2-x^2)q$ $xy''=y'-y'^{3}+A(1-y'^{2})^{\frac{3}{2}}$
    $L_{3;9}$ $X_1=(1+x^2)p+xyq, X_2=xyp+(1+y^2)q$,
    $X_3=yp-xq$ $y''=A[\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}]^{3/2}$
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    Table Ⅱ.  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 $K=\displaystyle{1+y'+(y-xy')^2\over 1+x^2+y^2}$
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