60.7.130 problem 1721 (book 6.130)

Internal problem ID [11719]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1721 (book 6.130)
Date solved : Monday, January 27, 2025 at 11:31:13 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime } y+a {y^{\prime }}^{2}+b y^{2} y^{\prime }+c y^{4}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 179

dsolve(diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)^2*diff(y(x),x)+c*y(x)^4=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ \left (2 a +4\right ) \left (\int _{}^{y}\frac {1}{\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} b \,\textit {\_a}^{2}-2 a \ln \left (\textit {\_a} \right ) \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}+2 \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\, \ln \left (2\right )-\sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\, \ln \left (\frac {\sec \left (\textit {\_Z} \right )^{2} \left (4 a c -b^{2}+8 c \right ) \textit {\_a}^{4}}{a +2}\right )+c_{1} \sqrt {\textit {\_a}^{4} \left (4 a c -b^{2}+8 c \right )}\right )\right ) \sqrt {\textit {\_a}^{4} \left (4 \left (a +2\right ) c -b^{2}\right )}-b \,\textit {\_a}^{2}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 50.939 (sec). Leaf size: 105

DSolve[c*y[x]^4 + b*y[x]^2*D[y[x],x] + a*D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{K[2]^2 \text {InverseFunction}\left [\frac {\log (c+\text {$\#$1} (b+(a+2) \text {$\#$1}))-\frac {2 b \arctan \left (\frac {b+2 (a+2) \text {$\#$1}}{\sqrt {4 (a+2) c-b^2}}\right )}{\sqrt {4 (a+2) c-b^2}}}{2 (a+2)}\&\right ][c_1-\log (K[2])]}dK[2]=x-c_2,y(x)\right ] \]