60.7.127 problem 1718 (book 6.127)

Internal problem ID [11716]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1718 (book 6.127)
Date solved : Monday, January 27, 2025 at 11:31:00 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.109 (sec). Leaf size: 109

dsolve(diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)^3=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ \left (2 a +3\right ) \left (\int _{}^{y}\frac {\textit {\_a}^{2 a}}{\sqrt {\left (2 a +3\right ) \left (-2 b \,\textit {\_a}^{4 a +3}+\textit {\_a}^{2 a} c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \left (-2 a -3\right ) \left (\int _{}^{y}\frac {\textit {\_a}^{2 a}}{\sqrt {\left (2 a +3\right ) \left (-2 b \,\textit {\_a}^{4 a +3}+\textit {\_a}^{2 a} c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 94.317 (sec). Leaf size: 277

DSolve[b*y[x]^3 + a*D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\frac {y(x) \sqrt {(2 a+3) y(x)^{2 a}} \sqrt {1-\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {a+1}{2 a+3},\frac {a+1}{2 a+3}+1,\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}\right )}{(a+1) \sqrt {-2 b y(x)^{2 a+3}+2 a c_1+3 c_1}}&=-x+c_2,y(x)\right ] \\ \text {Solve}\left [\frac {y(x) \sqrt {(2 a+3) y(x)^{2 a}} \sqrt {1-\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {a+1}{2 a+3},\frac {a+1}{2 a+3}+1,\frac {2 b y(x)^{2 a+3}}{2 a c_1+3 c_1}\right )}{(a+1) \sqrt {-2 b y(x)^{2 a+3}+2 a c_1+3 c_1}}&=x+c_2,y(x)\right ] \\ \end{align*}