Internal problem ID [10120]
Internal file name [OUTPUT/9067_Monday_June_06_2022_06_21_08_AM_86141387/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1799 (book 6.208).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (y^{\prime } x -y\right )^{3}=0} \]
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 a change of variables {x -> y(x), y(x) -> x} 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 -> 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 symmetries linear in x and y(x) trying differential order: 2; exact nonlinear trying 2nd order, integrating factor of the form mu(y) trying 2nd order, integrating factor of the form mu(x,y) trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case trying 2nd order, integrating factor of the form mu(y,y) trying differential order: 2; mu polynomial in y trying 2nd order, integrating factor of the form mu(x,y) differential order: 2; looking for linear symmetries differential order: 2; found: 1 linear symmetries. Trying reduction of order `, `2nd order, trying reduction of order with given symmetries:`[x, y]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 170
dsolve(x^3*y(x)^2*diff(diff(y(x),x),x)+(x+y(x))*(x*diff(y(x),x)-y(x))^3=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {i \sqrt {3}\, \operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}+i \sqrt {3}\, \operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}+\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}-2 c_{1} \operatorname {BesselY}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f} +\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}-2 \operatorname {BesselJ}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f}}{\textit {\_f}^{\frac {3}{2}} \left (\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} +\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right )\right )}d \textit {\_f} \right )+2 c_{2} \right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 36.551 (sec). Leaf size: 248
DSolve[(x + y[x])*(-y[x] + x*y'[x])^3 + x^3*y[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\int _1^{\frac {y(x)}{x}}\frac {i \sqrt {3} \sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )-2 \operatorname {BesselJ}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) K[2]-2 \operatorname {BesselY}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 K[2]+i \sqrt {3} \operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}}{\left (\operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1\right ) K[2]^{3/2}}dK[2]-2 \log (x)+2 c_2=0,y(x)\right ] \]