Internal problem ID [10034]
Internal file name [OUTPUT/8981_Monday_June_06_2022_06_07_27_AM_85794964/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1713 (book 6.122).
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.
\[ \boxed {y^{\prime \prime } y-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2}=0} \]
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 trying 2nd order, 2 integrating factors of the form mu(x,y) trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful `, `-> Computing symmetries using: way = HINT `, `2nd order, trying reduction of order with given symmetries:`[0, y]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-f(x)*y(x)*diff(y(x),x)-g(x)*y(x)^2=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{2} {\mathrm e}^{c_{1} \left (\int {\mathrm e}^{\int f \left (x \right )d x}d x \right )+\int {\mathrm e}^{\int f \left (x \right )d x} \left (\int {\mathrm e}^{-\left (\int f \left (x \right )d x \right )} g \left (x \right )d x \right )d x} \]
✓ Solution by Mathematica
Time used: 2.033 (sec). Leaf size: 61
DSolve[-(g[x]*y[x]^2) - f[x]*y[x]*y'[x] - y'[x]^2 + y[x]*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \exp \left (\int _1^x\exp \left (\int _1^{K[3]}f(K[1])dK[1]\right ) \left (c_1+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]\right )dK[3]\right ) \]