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: 0.406 (sec). Leaf size: 985
dsolve(x*y(x)+(x^2+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-1+\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right ) x^{2}}{2} \\ y \left (x \right ) &= \frac {\left (-1+\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right ) x^{2}}{2} \\ y \left (x \right ) &= \frac {\left (-1+\frac {\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{x^{2}}+\frac {x^{2}}{\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}\right ) x^{2}}{2} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}-x^{4}-2 x^{2} \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}-\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x^{2}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}-x^{4}-2 x^{2} \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}-\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x^{2}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}-x^{4}-2 x^{2} \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}-\left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {2}{3}}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+2 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}\right ) x^{2}}{4 \left (2 c_{1}^{2}-x^{6}+2 c_{1} \sqrt {-x^{6}+c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \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*}