Internal problem ID [1979]
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: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {\left (2 x +3 x^{2} y\right ) y^{\prime }+y+2 x y^{2}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 380
dsolve((2*x+3*x^2*y(x))*diff(y(x),x)+(y(x)+2*y(x)^2*x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {2 c_{1}^{2} 2^{\frac {1}{3}}-2 c_{1} \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}+2^{\frac {2}{3}} \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {2}{3}}}{6 c_{1} x \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2 \left (i \sqrt {3}-1\right ) c_{1}^{2} 2^{\frac {1}{3}}-\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) 2^{\frac {2}{3}}+4 c_{1} \right )}{12 \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} c_{1} x} \\ y \left (x \right ) &= \frac {-2 \left (1+i \sqrt {3}\right ) c_{1}^{2} 2^{\frac {1}{3}}+\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (\left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}}-4 c_{1} \right )}{12 \left (\left (3 \sqrt {\frac {-12 c_{1} +81 x}{x}}\, x -2 c_{1} +27 x \right ) c_{1}^{2}\right )^{\frac {1}{3}} c_{1} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 21.875 (sec). Leaf size: 380
DSolve[(2*x+3*x^2*y[x])*y'[x]+(y[x]+2*y[x]^2*x)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (\frac {2}{\sqrt [3]{\frac {3}{2} \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+\frac {27 c_1 x^4}{2}-x^3}}+\frac {2^{2/3} \sqrt [3]{3 \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+27 c_1 x^4-2 x^3}}{x^2}-\frac {2}{x}\right ) \\ y(x)\to \frac {1}{12} \left (-\frac {2 \left (1+i \sqrt {3}\right )}{\sqrt [3]{\frac {3}{2} \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+\frac {27 c_1 x^4}{2}-x^3}}+\frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{3 \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+27 c_1 x^4-2 x^3}}{x^2}-\frac {4}{x}\right ) \\ y(x)\to \frac {1}{12} \left (\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{\frac {3}{2} \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+\frac {27 c_1 x^4}{2}-x^3}}-\frac {2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{3 \sqrt {3} \sqrt {c_1 x^7 (-4+27 c_1 x)}+27 c_1 x^4-2 x^3}}{x^2}-\frac {4}{x}\right ) \\ \end{align*}