29.25.34 problem 731

Internal problem ID [5321]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 731
Date solved : Monday, January 27, 2025 at 11:10:55 AM
CAS classification : [[_homogeneous, `class G`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 130

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

\[ -\int _{\textit {\_b}}^{x}\frac {\sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}+\textit {\_a} \right )}d \textit {\_a} +\int _{}^{y \left (x \right )}\frac {\textit {\_f}^{2} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right ) \left (\int _{\textit {\_b}}^{x}\frac {1}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )^{2}}d \textit {\_a} \right )-x -\sqrt {\textit {\_f}^{2}+x^{2}}}{\textit {\_f} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right )}d \textit {\_f} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.225 (sec). Leaf size: 1457

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

\begin{align*} y(x)\to -\frac {1}{2} \sqrt {\frac {x^6-x^4 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}+x^2 \left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}\right ){}^{2/3}+8 e^{6 c_1}}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {x^6-x^4 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}+x^2 \left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}\right ){}^{2/3}+8 e^{6 c_1}}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}}} \\ y(x)\to -\frac {\sqrt {\frac {i \left (\left (\sqrt {3}+i\right ) x^6+2 i x^4 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}-\left (\sqrt {3}-i\right ) x^2 \left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}\right ){}^{2/3}+8 \left (\sqrt {3}+i\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {i \left (\left (\sqrt {3}+i\right ) x^6+2 i x^4 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}-\left (\sqrt {3}-i\right ) x^2 \left (\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}\right ){}^{2/3}+8 \left (\sqrt {3}+i\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {i \left (x^2 \left (x^2+\sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}-\left (\sqrt {3}-i\right ) x^2\right )-8 \left (\sqrt {3}-i\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {i \left (x^2 \left (x^2+\sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}-\left (\sqrt {3}-i\right ) x^2\right )-8 \left (\sqrt {3}-i\right ) e^{6 c_1}\right )}{x^2 \sqrt [3]{\frac {-x^{12}+20 e^{6 c_1} x^6+8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}+8 e^{12 c_1}}{x^6}}}}}{2 \sqrt {2}} \\ \end{align*}