Internal problem ID [10031]
Internal file name [OUTPUT/8978_Monday_June_06_2022_06_06_56_AM_11503740/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1710 (book 6.119).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}-\left (a y-1\right ) y^{\prime }+2 a^{2} y^{2}-2 b^{2} y^{3}+a y=0} \]
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right ) y +\left (-a y -p \left (y \right )+1\right ) p \left (y \right )+\left (-2 y^{2} b^{2}+2 y \,a^{2}+a \right ) y = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\). Unable to determine ODE type.
Unable to solve. Terminating
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-(2*_a^3*b^2-2*_a^2*a^2+_a*_b(_a)*a+_b(_a)^2-a*_a-_b(_a))/_a = 0, _b(_a)` Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel Looking for potential symmetries Looking for potential symmetries Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 2 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> trying a symmetry pattern of the form [F(x)+G(y), 0] -> trying a symmetry pattern of the form [0, F(x)+G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying a symmetry pattern of conformal type -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y) trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying differential order: 2; exact nonlinear trying 2nd order, integrating factor of the form mu(x,y) -> trying 2nd order, the S-function method -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for the S-function -> trying 2nd order, the S-function method -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V -> trying 2nd order, No Point Symmetries Class V --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, No Point Symmetries Class V trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^ --- trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods for dynamical symmetries --- -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x,y)* --- Trying Lie symmetry methods, 2nd order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5 `, `-> Computing symmetries using: way = formal *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✗ Solution by Maple
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-(a*y(x)-1)*diff(y(x),x)+2*a^2*y(x)^2-2*b^2*y(x)^3+a*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 114.511 (sec). Leaf size: 540
DSolve[a*y[x] + 2*a^2*y[x]^2 - 2*b^2*y[x]^3 - (-1 + a*y[x])*y'[x] - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {1}{2 a}+e^{2 a x} \left (\frac {e^{-2 a x} \left (c_1 \left (a^{3/2}-\sqrt {a^3+2 b^2}\right ) \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \sqrt {a b^2 c_2 e^{2 a x}} \operatorname {BesselJ}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\operatorname {Gamma}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) \left (\left (a^{3/2}+\sqrt {a^3+2 b^2}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 \sqrt {a b^2 c_2 e^{2 a x}} \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1,\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right )\right ){}^2}{4 a b^2 \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\operatorname {Gamma}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right ){}^2}+c_2\right ) \]