Internal problem ID [3469]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 731.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _dAlembert]
Solve \begin {gather*} \boxed {x \left (x +\sqrt {x^{2}+y^{2}}\right ) y^{\prime }+y \sqrt {x^{2}+y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 139
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)
\[ \int _{\textit {\_b}}^{x}-\frac {\sqrt {\textit {\_a}^{2}+y \relax (x )^{2}}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+y \relax (x )^{2}}+\textit {\_a} \right )}d \textit {\_a} +\int _{}^{y \relax (x )}\left (-\frac {\sqrt {\textit {\_f}^{2}+x^{2}}+x}{\textit {\_f} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right )}-\left (\int _{\textit {\_b}}^{x}\left (-\frac {\textit {\_f}}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )}+\frac {2 \textit {\_f}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )^{2}}\right )d \textit {\_a} \right )\right )d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.213 (sec). Leaf size: 1435
DSolve[x(x+Sqrt[x^2+y[x]^2])y'[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]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}+x^2 \left (-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}\right ){}^{2/3}+8 e^{6 c_1}}{x^2 \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}} \\ y(x)\to \frac {1}{2} \sqrt {\frac {x^6-x^4 \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}+x^2 \left (-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}\right ){}^{2/3}+8 e^{6 c_1}}{x^2 \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}} \\ y(x)\to -\frac {\sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}+\frac {8 i \left (\sqrt {3}+i\right ) e^{6 c_1}}{x^2 \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\left (-1-i \sqrt {3}\right ) \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}-2 x^2+\frac {i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}+\frac {8 i \left (\sqrt {3}+i\right ) e^{6 c_1}}{x^2 \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}}}{2 \sqrt {2}} \\ y(x)\to -\frac {\sqrt {\frac {i x^2 \left (x^2+\sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}-\left (\sqrt {3}-i\right ) x^2\right )+\left (-8-8 i \sqrt {3}\right ) e^{6 c_1}}{x^2 \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}}}{2 \sqrt {2}} \\ y(x)\to \frac {\sqrt {\frac {i x^2 \left (x^2+\sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}\right ) \left (\left (\sqrt {3}+i\right ) \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}-\left (\sqrt {3}-i\right ) x^2\right )+\left (-8-8 i \sqrt {3}\right ) e^{6 c_1}}{x^2 \sqrt [3]{-x^6+\frac {8 e^{12 c_1}}{x^6}+\frac {8 \sqrt {e^{6 c_1} \left (-x^6+e^{6 c_1}\right ){}^3}}{x^6}+20 e^{6 c_1}}}}}{2 \sqrt {2}} \\ \end{align*}