60.7.122 problem 1713 (book 6.122)

Internal problem ID [11711]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1713 (book 6.122)
Date solved : Tuesday, January 28, 2025 at 06:07:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime } y-{y^{\prime }}^{2}-f \left (x \right ) y y^{\prime }-g \left (x \right ) y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.063 (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)
 

\[ y = c_{2} {\mathrm e}^{c_{1} \left (\int {\mathrm e}^{\int fd x}d x \right )+\int {\mathrm e}^{\int fd x} \left (\int {\mathrm e}^{-\int fd x} g \left (x \right )d x \right )d x} \]

✓ Solution by Mathematica

Time used: 1.187 (sec). Leaf size: 61

DSolve[-(g[x]*y[x]^2) - f[x]*y[x]*D[y[x],x] - D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\exp \left (\int _1^{K[3]}f(K[1])dK[1]\right ) \left (c_1+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]\right )dK[3]\right ) \]