62.15.1 problem Ex 1

Internal problem ID [12888]
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
Date solved : Tuesday, January 28, 2025 at 04:31:55 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x +y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.184 (sec). Leaf size: 768

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

\begin{align*} y &= -\frac {i \sqrt {2}\, x}{2} \\ y &= \frac {i \sqrt {2}\, x}{2} \\ y &= \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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{1}/{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\ y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{1}/{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{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 )^{{3}/{2}}}\right )}^{{1}/{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[x+D[y[x],x]*y[x]*(2*(D[y[x],x])^2+3)==0,y[x],x,IncludeSingularSolutions -> True]
 

Timed out