Internal problem ID [3977]
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]
\[ \boxed {x \left (x +\sqrt {y^{2}+x^{2}}\right ) y^{\prime }+\sqrt {y^{2}+x^{2}}\, y=0} \]
✓ Solution by Maple
Time used: 0.031 (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)
\[ -\left (\int _{\textit {\_b}}^{x}\frac {\sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}+\textit {\_a} \right )}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {\textit {\_f}^{2} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right ) \left (\int _{\textit {\_b}}^{x}\frac {1}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )^{2}}d \textit {\_a} \right )-x -\sqrt {\textit {\_f}^{2}+x^{2}}}{\textit {\_f} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right )}d \textit {\_f} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.36 (sec). Leaf size: 1457
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]{\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*}