Internal problem ID [8583]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 247.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {\left (3 x +2\right ) \left (y-2 x -1\right ) y^{\prime }-y^{2}+y x=7 x^{2}+9 x +3} \]
✓ Solution by Maple
Time used: 0.329 (sec). Leaf size: 517
dsolve((3*x+2)*(y(x)-2*x-1)*diff(y(x),x)-y(x)^2+x*y(x)-7*x^2-9*x-3=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {1}{3}+\frac {\left (3 x +2\right ) \left (7 {\left (-\frac {\left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {9 \left (3 x +2\right )^{2} c_{1}^{2}}{4 \left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {3 \left (3 x +2\right ) c_{1}}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {9 \left (3 x +2\right )^{2} c_{1}^{2}}{2 \left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}-1\right )}{6 {\left (-\frac {\left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {9 \left (3 x +2\right )^{2} c_{1}^{2}}{4 \left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {3 \left (3 x +2\right ) c_{1}}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {9 \left (3 x +2\right )^{2} c_{1}^{2}}{2 \left (2 \left (3 x +2\right ) c_{1} -27 \left (3 x +2\right )^{3} c_{1}^{3}+2 \sqrt {-27 \left (3 x +2\right )^{4} c_{1}^{4}+\left (3 x +2\right )^{2} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )}^{2}} \]
✓ Solution by Mathematica
Time used: 66.883 (sec). Leaf size: 590
DSolve[(3*x+2)*(y[x]-2*x-1)*y'[x]-y[x]^2+x*y[x]-7*x^2-9*x-3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {9 x^2+x \left (12+\sqrt [3]{27 x^3+54 x^2+36 x-2 e^{2 c_1} (3 x+2)+2 \sqrt {e^{2 c_1} (3 x+2)^2 \left (-(3 x+2)^2+e^{2 c_1}\right )}+8}\right )+\left (27 x^3+54 x^2+36 x-2 e^{2 c_1} (3 x+2)+2 \sqrt {e^{2 c_1} (3 x+2)^2 \left (-(3 x+2)^2+e^{2 c_1}\right )}+8\right ){}^{2/3}+4}{2 \sqrt [3]{27 x^3+54 x^2+36 x-2 e^{2 c_1} (3 x+2)+2 \sqrt {e^{2 c_1} (3 x+2)^2 \left (-(3 x+2)^2+e^{2 c_1}\right )}+8}} y(x)\to -\frac {i \left (\sqrt {3}-i\right ) (3 x+2)^2}{4 \sqrt [3]{27 x^3+54 x^2+36 x-2 e^{2 c_1} (3 x+2)+2 \sqrt {e^{2 c_1} (3 x+2)^2 \left (-(3 x+2)^2+e^{2 c_1}\right )}+8}}+\frac {1}{4} i \left (\sqrt {3}+i\right ) \sqrt [3]{27 x^3+54 x^2+36 x-2 e^{2 c_1} (3 x+2)+2 \sqrt {e^{2 c_1} (3 x+2)^2 \left (-(3 x+2)^2+e^{2 c_1}\right )}+8}+\frac {x}{2} y(x)\to \frac {i \left (\sqrt {3}+i\right ) (3 x+2)^2}{4 \sqrt [3]{27 x^3+54 x^2+36 x-2 e^{2 c_1} (3 x+2)+2 \sqrt {e^{2 c_1} (3 x+2)^2 \left (-(3 x+2)^2+e^{2 c_1}\right )}+8}}-\frac {1}{4} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^3+54 x^2+36 x-2 e^{2 c_1} (3 x+2)+2 \sqrt {e^{2 c_1} (3 x+2)^2 \left (-(3 x+2)^2+e^{2 c_1}\right )}+8}+\frac {x}{2} \end{align*}