Internal problem ID [9991]
Internal file name [OUTPUT/8938_Monday_June_06_2022_06_00_26_AM_91579514/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1670 (book 6.79).
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_xy]]
Unable to solve or complete the solution.
\[ \boxed {x y^{\prime \prime }-x^{2} {y^{\prime }}^{2}+2 y^{\prime }+y^{2}=0} \]
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 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 integrating factor(s) found: y(x)^2*exp(-y(x)*x)/diff(y(x),x)^2 attempting the computation of a related first integral... -> Calling odsolve with the ODE`, _a^2*exp(-_b(_a)*_a)*(diff(_b(_a), _a))-_b(_a)*exp(-_b(_a)*_a)*_a-exp(-_b(_a)*_a)+c__1 = 0, _b( 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 homogeneous G <- homogeneous successful <- differential order: 2; exact nonlinear successful <- 2nd_order mu_xyp successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 32
dsolve(x*diff(diff(y(x),x),x)-x^2*diff(y(x),x)^2+2*diff(y(x),x)+y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{2} -\left (\int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_f} -1+{\mathrm e}^{\textit {\_f}} c_{1}}d \textit {\_f} \right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.8 (sec). Leaf size: 160
DSolve[y[x]^2 + 2*y'[x] - x^2*y'[x]^2 + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x}{e^{x K[1]} c_1+2 x K[1]+1}dK[1]-\int _1^x\left (\int _1^{y(x)}\left (\frac {\left (e^{K[1] K[2]} c_1 K[1]+2 K[1]\right ) K[2]}{\left (e^{K[1] K[2]} c_1+2 K[1] K[2]+1\right ){}^2}-\frac {1}{e^{K[1] K[2]} c_1+2 K[1] K[2]+1}\right )dK[1]-\frac {e^{K[2] y(x)} c_1+K[2] y(x)+1}{K[2] \left (e^{K[2] y(x)} c_1+2 K[2] y(x)+1\right )}\right )dK[2]=c_2,y(x)\right ] \]