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: 0.485 (sec). Leaf size: 392
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 {\frac {14 \left (x +\frac {2}{3}\right ) \left (-\frac {4}{1701}+\left (x^{2}+\frac {26}{21} x +\frac {8}{21}\right ) c_{1}^{2}\right ) {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {2}{3}}}{27}+21 \left (\frac {\left (-\frac {2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}}{2187}+\left (-\frac {2}{729}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right ) c_{1} \right ) \left (1+i \sqrt {3}\right ) {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {1}{3}}}{9}+\left (i \sqrt {3}-1\right ) \left (x +\frac {2}{3}\right ) \left (-\frac {4 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}}{2187}+\left (-\frac {2}{27}+\left (x +\frac {2}{3}\right ) c_{1} \right ) \left (x +\frac {2}{3}\right ) \left (\frac {2}{27}+\left (x +\frac {2}{3}\right ) c_{1} \right ) c_{1} \right ) c_{1} \right ) \left (x +\frac {4}{7}\right )}{{\left (\frac {\left (1-i \sqrt {3}\right ) {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {2}{3}}}{81}+\left (x +\frac {2}{3}\right ) \left (\frac {2 {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {1}{3}}}{9}+\left (x +\frac {2}{3}\right ) c_{1} \left (1+i \sqrt {3}\right )\right ) c_{1} \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*}