Internal problem ID [10194]
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]
\[ \boxed {x +y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right )=0} \]
✓ Solution by Maple
Time used: 0.532 (sec). Leaf size: 776
dsolve(x+diff(y(x),x)*y(x)*(2*diff(y(x),x)^2+3)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {i \sqrt {2}\, x}{2} \\ y \left (x \right ) = \frac {i \sqrt {2}\, x}{2} \\ y \left (x \right ) = \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {-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}+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}} \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}}+\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} +c_{1} \right ) x \\ y \left (x \right ) = \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {2 i {\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}} \sqrt {3}\, \textit {\_a}^{2}+i {\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}} \sqrt {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 \left (x \right ) = \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 i {\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}} \sqrt {3}\, \textit {\_a}^{2}+i {\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}} \sqrt {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