Internal problem ID [10198]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first. Article
26. Equations solvable for \(x\). Page 55
Problem number: Ex 1.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [[_homogeneous, class A], _dAlembert]
Solve \begin {gather*} \boxed {x +y^{\prime } y \left (2 \left (y^{\prime }\right )^{2}+3\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 822
dsolve(x+diff(y(x),x)*y(x)*(2*diff(y(x),x)^2+3)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {i \sqrt {2}\, x}{2} \\ y \relax (x ) = \frac {i \sqrt {2}\, x}{2} \\ y \relax (x ) = \RootOf \left (-\ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {2 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}-2 \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}} \textit {\_a}^{3}+\left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {2}{3}}-\textit {\_a} \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}}-\textit {\_a}^{2}}{\left (\textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{2}+1\right ) \left (-\frac {\textit {\_a} \left (-2 \textit {\_a}^{4}+\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}-3 \textit {\_a}^{2}-1\right )}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {5}{2}}}\right )^{\frac {1}{3}}}d \textit {\_a} +c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-2 \ln \relax (x )+\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}+i \sqrt {3}\, \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}+i \sqrt {3}\, \textit {\_a}^{2}-2 \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}-4 \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}} \textit {\_a}^{3}-\left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}+\textit {\_a}^{2}}{\left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1}\right ) x \\ y \relax (x ) = \RootOf \left (-2 \ln \relax (x )-\left (\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}+i \sqrt {3}\, \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}+i \sqrt {3}\, \textit {\_a}^{2}+2 \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}} \textit {\_a}^{2}+4 \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}} \textit {\_a}^{3}+\left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}}-\textit {\_a}^{2}}{\left (-\frac {\left (-\textit {\_a}^{2}+\sqrt {2 \textit {\_a}^{2}+1}-1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{\frac {3}{2}}}\right )^{\frac {1}{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+2 c_{1}\right ) x \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[x+y'[x]*y[x]*(2*(y'[x])^2+3)==0,y[x],x,IncludeSingularSolutions -> True]
Timed out