Internal problem ID [8607]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 271.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (2 x +y\right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 417
dsolve((y(x)^2+x^2)*diff(y(x),x)+2*x*(y(x)+2*x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\frac {\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}-\frac {2 c_{1} x^{2}}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}}{\sqrt {c_{1}}} y \left (x \right ) = \frac {-\frac {\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {c_{1} x^{2}}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1} x^{2}}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} y \left (x \right ) = \frac {-\frac {\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4}+\frac {c_{1} x^{2}}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1} x^{2}}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 c_{1}^{3} x^{6}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}\right )}{2}}{\sqrt {c_{1}}} \end{align*}
✓ Solution by Mathematica
Time used: 19.146 (sec). Leaf size: 593
DSolve[(y[x]^2+x^2)*y'[x]+2*x*(y[x]+2*x)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {20 x^6-8 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} y(x)\to \sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}-\frac {x^2}{\sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2+\left (-1-i \sqrt {3}\right ) \left (\sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} y(x)\to \frac {\left (1+i \sqrt {3}\right ) x^2+i \left (\sqrt {3}+i\right ) \left (\sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-2 x^3}} \end{align*}