Internal problem ID [1967]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 9, page 38
Problem number: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class A`]]
\[ \boxed {y x +\left (x^{2}+y\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 1153
dsolve(x*y(x)+(x^2+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y = \frac {\left (\frac {4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {4 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4\right ) x^{2}}{8} y = \frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (\frac {4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {4 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4\right ) x^{2}}{8} y = \frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (\frac {4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {4 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4\right ) x^{2}}{8} y = \frac {\left (-\frac {2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}-\frac {2 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4-4 i \sqrt {3}\, \left (\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {x^{2}}{2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{8} y = \frac {\left (-\frac {2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}-\frac {2 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4+4 i \sqrt {3}\, \left (\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {x^{2}}{2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{8} y = \frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (-\frac {2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}-\frac {2 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4-4 i \sqrt {3}\, \left (\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {x^{2}}{2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{8} y = \frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{3} \left (-\frac {2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}-\frac {2 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4+4 i \sqrt {3}\, \left (\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {x^{2}}{2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{8} y = \frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (-\frac {2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}-\frac {2 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4-4 i \sqrt {3}\, \left (\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {x^{2}}{2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{8} y = \frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{3} \left (-\frac {2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}-\frac {2 x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}-4+4 i \sqrt {3}\, \left (\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2 x^{2}}-\frac {x^{2}}{2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{8} \end{align*}
✓ Solution by Mathematica
Time used: 60.038 (sec). Leaf size: 397
DSolve[x*y[x]+(x^2+y[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,1\right ]} y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,2\right ]} y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,3\right ]} y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,4\right ]} y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,5\right ]} y(x)\to -x^2+\frac {1}{\text {Root}\left [\text {$\#$1}^6 \left (x^{12}+e^{12 c_1}\right )-6 \text {$\#$1}^4 x^8+4 \text {$\#$1}^3 x^6+9 \text {$\#$1}^2 x^4-12 \text {$\#$1} x^2+4\&,6\right ]} \end{align*}