Internal problem ID [10000]
Internal file name [OUTPUT/8947_Monday_June_06_2022_06_01_52_AM_68787660/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1679 (book 6.88).
ODE order: 2.
ODE degree: 2.
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.
\[ \boxed {x^{2} y^{\prime \prime }-\sqrt {a \,x^{2} {y^{\prime }}^{2}+y^{2} b}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- 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 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 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases trying symmetries linear in x and y(x) <- linear symmetries successful`
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 64
dsolve(x^2*diff(diff(y(x),x),x)-(a*x^2*diff(y(x),x)^2+y(x)^2*b)^(1/2)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right )-{\mathrm e}^{\int _{}^{\ln \left (x \right )}\operatorname {RootOf}\left (-y \left (x \right ) \left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a}^{2} y \left (x \right )-\textit {\_a} y \left (x \right )-\sqrt {y \left (x \right )^{2} \left (a \,\textit {\_a}^{2}+b \right )}}d \textit {\_a} \right )-\textit {\_b} +c_{1} \right )d \textit {\_b} +c_{2}} &= 0 \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[-Sqrt[b*y[x]^2 + a*x^2*y'[x]^2] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
Not solved