2.105   ODE No. 105

\[ a x y(x)^2+b y(x)+c x+d+x y'(x)=0 \]

✓ Mathematica : cpu = 0.14705 (sec), leaf count = 473

DSolve[d + c*x + b*y[x] + a*x*y[x]^2 + x*Derivative[1][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to \frac {c_1 \left (i \sqrt {a} e^{-i \sqrt {a} \sqrt {c} x} \left (b \left (-\sqrt {c}\right )-i \sqrt {a} d\right ) U\left (1-\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}},b+1,2 i \sqrt {a} \sqrt {c} x\right )-i \sqrt {a} \sqrt {c} e^{-i \sqrt {a} \sqrt {c} x} U\left (-\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}},b,2 i \sqrt {a} \sqrt {c} x\right )\right )-i \sqrt {a} \sqrt {c} e^{-i \sqrt {a} \sqrt {c} x} L_{\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )-2 i \sqrt {a} \sqrt {c} e^{-i \sqrt {a} \sqrt {c} x} L_{\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}}-1}^b\left (2 i \sqrt {a} \sqrt {c} x\right )}{a \left (c_1 e^{-i \sqrt {a} \sqrt {c} x} U\left (-\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}},b,2 i \sqrt {a} \sqrt {c} x\right )+e^{-i \sqrt {a} \sqrt {c} x} L_{\frac {-\sqrt {c} b-i \sqrt {a} d}{2 \sqrt {c}}}^{b-1}\left (2 i \sqrt {a} \sqrt {c} x\right )\right )}\right \}\right \}\]

✓ Maple : cpu = 0.421 (sec), leaf count = 844

dsolve(x*diff(y(x),x)+a*x*y(x)^2+b*y(x)+c*x+d = 0,y(x))
 

\[y \left (x \right ) = -\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 )^{{3}/{2}} a b c d -2 d b \left (-a c \right )^{{5}/{2}}\right ) \operatorname {KummerU}\left (\frac {\left (-a c \right )^{{3}/{2}} d +\left (2 \sqrt {-a c}\, d +c \left (b +2\right )\right ) c a}{2 c^{2} a}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right )}{4}+\left (a \,c^{3} \left (a d -b \sqrt {-a c}\right ) \operatorname {KummerM}\left (\frac {\left (-a c \right )^{{3}/{2}} d +\left (2 \sqrt {-a c}\, d +c \left (b +2\right )\right ) c a}{2 c^{2} a}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right )+a \,c^{3} \left (b \sqrt {-a c}+a d \right ) \operatorname {KummerM}\left (\frac {\left (-a c \right )^{{3}/{2}} d +2 \sqrt {-a c}\, a d c +a \,c^{2} b}{2 a \,c^{2}}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right )-\frac {\operatorname {KummerU}\left (\frac {\left (-a c \right )^{{3}/{2}} d +2 \sqrt {-a c}\, a d c +a \,c^{2} b}{2 a \,c^{2}}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right ) c_{1} \left (b c -\sqrt {-a c}\, d \right )}{2}\right ) c^{2} a^{2}\right )}{-c_{1} \left (a^{2} b^{2} c^{4} \sqrt {-a c}+2 a c \,d^{2} \left (-a c \right )^{{5}/{2}}+\left (-a c \right )^{{7}/{2}} d^{2}\right ) \operatorname {KummerU}\left (\frac {\left (-a c \right )^{{3}/{2}} d +\left (2 \sqrt {-a c}\, d +c \left (b +2\right )\right ) c a}{2 c^{2} a}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right )+4 c^{4} a^{2} \left (a^{2} c^{2} \left (\sqrt {-a c}\, d +b c \right ) \operatorname {KummerM}\left (\frac {\left (-a c \right )^{{3}/{2}} d +\left (2 \sqrt {-a c}\, d +c \left (b +2\right )\right ) c a}{2 c^{2} a}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right )+a^{2} c^{2} \left (b c -\sqrt {-a c}\, d \right ) \operatorname {KummerM}\left (\frac {\left (-a c \right )^{{3}/{2}} d +2 \sqrt {-a c}\, a d c +a \,c^{2} b}{2 a \,c^{2}}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right )+\frac {\operatorname {KummerU}\left (\frac {\left (-a c \right )^{{3}/{2}} d +2 \sqrt {-a c}\, a d c +a \,c^{2} b}{2 a \,c^{2}}, \frac {\left (-a c \right )^{{3}/{2}} d +\left (\sqrt {-a c}\, d +c \left (b +1\right )\right ) c a}{c^{2} a}, 2 x \sqrt {-a c}\right ) c_{1} \left (b \sqrt {-a c}+a d \right )}{2}\right )}\]