Internal problem ID [10095]
Internal file name [OUTPUT/9042_Monday_June_06_2022_06_16_28_AM_66370367/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1774 (book 6.183).
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.
\[ \boxed {2 x^{2} y^{\prime \prime } y-x^{2} \left ({y^{\prime }}^{2}+1\right )+y^{2}=0} \]
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 2nd order ODE linearizable_by_differentiation -> Calling odsolve with the ODE`, diff(diff(diff(y(x), x), x), x)-(-(diff(y(x), x))*x+y(x))/x^3, y(x)` *** Sublevel 2 *** Methods for third order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type <- LODE of Euler type successful <- 2nd order ODE linearizable_by_differentiation successful`
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 30
dsolve(2*x^2*y(x)*diff(diff(y(x),x),x)-x^2*(diff(y(x),x)^2+1)+y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x \left (4 c_{2}^{2} \ln \left (x \right )^{2}+4 c_{1} \ln \left (x \right ) c_{2} +c_{1}^{2}+1\right )}{4 c_{2}} \]
✓ Solution by Mathematica
Time used: 0.845 (sec). Leaf size: 49
DSolve[y[x]^2 - x^2*(1 + y'[x]^2) + 2*x^2*y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (c_1{}^2 \log ^2(x)-2 c_2 c_1{}^2 \log (x)+4+c_2{}^2 c_1{}^2\right )}{4 c_1} \\ y(x)\to \text {Indeterminate} \\ \end{align*}