Internal problem ID [7686]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 105.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x y^{\prime }+a x y^{2}+b y+c x +d=0} \end {gather*}
✓ Solution by Maple
Time used: 0.021 (sec). Leaf size: 844
dsolve(x*diff(y(x),x) + a*x*y(x)^2 + b*y(x) + c*x + d=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {4 c^{2} \left (-\frac {c_{1} \left (a^{3} c^{2} d^{2}+a^{2} b^{2} c^{3}-2 \left (-a c \right )^{\frac {3}{2}} a b c d -2 d b \left (-a c \right )^{\frac {5}{2}}\right ) \KummerU \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (2 \sqrt {-a c}\, d +c \left (2+b \right )\right ) a}{2 a \,c^{2}}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right )}{4}+c^{2} \left (a \,c^{3} \left (a d -b \sqrt {-a c}\right ) \KummerM \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (2 \sqrt {-a c}\, d +c \left (2+b \right )\right ) a}{2 a \,c^{2}}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right )+a \,c^{3} \left (b \sqrt {-a c}+a d \right ) \KummerM \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +2 a d c \sqrt {-a c}+a b \,c^{2}}{2 c^{2} a}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right )-\frac {\KummerU \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +2 a d c \sqrt {-a c}+a b \,c^{2}}{2 c^{2} a}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right ) c_{1} \left (c b -\sqrt {-a c}\, d \right )}{2}\right ) a^{2}\right )}{-c_{1} \left (a^{2} b^{2} c^{4} \sqrt {-a c}+2 a c \,d^{2} \left (-a c \right )^{\frac {5}{2}}+\left (-a c \right )^{\frac {7}{2}} d^{2}\right ) \KummerU \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (2 \sqrt {-a c}\, d +c \left (2+b \right )\right ) a}{2 a \,c^{2}}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right )+4 c^{4} a^{2} \left (a^{2} c^{2} \left (\sqrt {-a c}\, d +c b \right ) \KummerM \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (2 \sqrt {-a c}\, d +c \left (2+b \right )\right ) a}{2 a \,c^{2}}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right )+a^{2} c^{2} \left (c b -\sqrt {-a c}\, d \right ) \KummerM \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +2 a d c \sqrt {-a c}+a b \,c^{2}}{2 c^{2} a}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right )+\frac {\KummerU \left (\frac {\left (-a c \right )^{\frac {3}{2}} d +2 a d c \sqrt {-a c}+a b \,c^{2}}{2 c^{2} a}, \frac {\left (-a c \right )^{\frac {3}{2}} d +c \left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) a}{a \,c^{2}}, 2 x \sqrt {-a c}\right ) c_{1} \left (b \sqrt {-a c}+a d \right )}{2}\right )} \]
✓ Solution by Mathematica
Time used: 0.446 (sec). Leaf size: 421
DSolve[x*y'[x] + a*x*y[x]^2 + b*y[x] + c*x + d==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \left (\sqrt {c} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )+c_1 \left (b \sqrt {c}+i \sqrt {a} d\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {a} d}{\sqrt {c}}+b+2\right ),b+1,2 i \sqrt {a} \sqrt {c} x\right )+\sqrt {c} \left (2 L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}-1}^b\left (2 i \sqrt {a} \sqrt {c} x\right )+L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )\right )}{\sqrt {a} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )+L_{-\frac {b}{2}-\frac {i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )} \\ y(x)\to \frac {\frac {\left (\sqrt {a} d-i b \sqrt {c}\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {a} d}{\sqrt {c}}+b+2\right ),b+1,2 i \sqrt {a} \sqrt {c} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (b+\frac {i \sqrt {a} d}{\sqrt {c}}\right ),b,2 i \sqrt {a} \sqrt {c} x\right )}-i \sqrt {c}}{\sqrt {a}} \\ \end{align*}