Internal problem ID [10106]
Internal file name [OUTPUT/9053_Monday_June_06_2022_06_17_46_AM_2217187/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1785 (book 6.194).
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], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Unable to solve or complete the solution.
\[ \boxed {\left (y^{2}+x^{2}\right ) y^{\prime \prime }-\left ({y^{\prime }}^{2}+1\right ) \left (y^{\prime } x -y\right )=0} \]
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) -> Calling odsolve with the ODE`, arctan(diff(_b(_a), _a))-arctan(_b(_a)/_a)+c__1 = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: -> Solving 1st order ODE of high degree, 1st attempt trying 1st order WeierstrassP solution for high degree ODE trying 1st order WeierstrassPPrime solution for high degree ODE trying 1st order JacobiSN solution for high degree ODE trying 1st order ODE linearizable_by_differentiation trying differential order: 1; missing variables trying homogeneous B trying homogeneous types: trying homogeneous D <- homogeneous successful <- differential order: 2; exact nonlinear successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 57
dsolve((y(x)^2+x^2)*diff(diff(y(x),x),x)-(diff(y(x),x)^2+1)*(x*diff(y(x),x)-y(x))=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (\cos \left (\textit {\_Z} \right )^{2} {\mathrm e}^{-\frac {2 \left (\textit {\_Z} c_{1} i+i \textit {\_Z} +c_{2} c_{1} -c_{2} \right )}{-1+c_{1}}}-x^{2}\right )\right ) x \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.322 (sec). Leaf size: 74
DSolve[(y[x] - x*y'[x])*(1 + y'[x]^2) + (x^2 + y[x]^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \left (\log \left (1-\frac {i y(x)}{x}\right )+\log \left (1+\frac {i y(x)}{x}\right )+i \cot (c_1) \left (\log \left (1-\frac {i y(x)}{x}\right )-\log \left (1+\frac {i y(x)}{x}\right )\right )\right )=-\log (x)+c_2,y(x)\right ] \]