##### 4.4.32 $$\text {a0}+\text {a1} x+y(x) (\text {a2}+\text {a3} x y(x))+x y'(x)=0$$

ODE
$\text {a0}+\text {a1} x+y(x) (\text {a2}+\text {a3} x y(x))+x y'(x)=0$ ODE Classiﬁcation

[_rational, _Riccati]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.417424 (sec), leaf count = 301

$\left \{\left \{y(x)\to -\frac {i \left (\sqrt {\text {a1}} c_1 U\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+c_1 \left (\sqrt {\text {a1}} \text {a2}+i \text {a0} \sqrt {\text {a3}}\right ) U\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+\sqrt {\text {a1}} \left (2 L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}-1}^{\text {a2}}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )\right )}{\sqrt {\text {a3}} \left (c_1 U\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )}\right \}\right \}$

Maple
cpu = 0.241 (sec), leaf count = 848

$\left [y \left (x \right ) = \frac {4 \mathit {a1}^{2} \left (-\frac {\textit {\_C1} \left (\mathit {a0}^{2} \mathit {a1}^{2} \mathit {a3}^{3}+\mathit {a1}^{3} \mathit {a2}^{2} \mathit {a3}^{2}-2 \left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} \mathit {a1} \mathit {a2} \mathit {a3} -2 \left (-\mathit {a1} \mathit {a3} \right )^{\frac {5}{2}} \mathit {a0} \mathit {a2} \right ) \KummerU \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\frac {\mathit {a1} \left (\mathit {a2} +2\right )}{2}\right )}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right )}{4}+\left (\mathit {a1}^{3} \mathit {a3} \left (\mathit {a3} \mathit {a0} -\mathit {a2} \sqrt {-\mathit {a1} \mathit {a3}}\right ) \KummerM \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\frac {\mathit {a1} \left (\mathit {a2} +2\right )}{2}\right )}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right )+\mathit {a1}^{3} \mathit {a3} \left (\mathit {a2} \sqrt {-\mathit {a1} \mathit {a3}}+\mathit {a3} \mathit {a0} \right ) \KummerM \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a0} \mathit {a3} \mathit {a1} \sqrt {-\mathit {a1} \mathit {a3}}+\mathit {a1}^{2} \mathit {a2} \mathit {a3}}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right )+\frac {\KummerU \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a0} \mathit {a3} \mathit {a1} \sqrt {-\mathit {a1} \mathit {a3}}+\mathit {a1}^{2} \mathit {a2} \mathit {a3}}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right ) \textit {\_C1} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} -\mathit {a1} \mathit {a2} \right )}{2}\right ) \mathit {a3}^{2} \mathit {a1}^{2}\right )}{\textit {\_C1} \left (\mathit {a1}^{4} \mathit {a2}^{2} \mathit {a3}^{2} \sqrt {-\mathit {a1} \mathit {a3}}+2 \mathit {a0}^{2} \mathit {a1} \mathit {a3} \left (-\mathit {a1} \mathit {a3} \right )^{\frac {5}{2}}+\left (-\mathit {a1} \mathit {a3} \right )^{\frac {7}{2}} \mathit {a0}^{2}\right ) \KummerU \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\frac {\mathit {a1} \left (\mathit {a2} +2\right )}{2}\right )}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right )+4 \mathit {a3}^{2} \left (-\mathit {a1}^{2} \mathit {a3}^{2} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \mathit {a2} \right ) \KummerM \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\frac {\mathit {a1} \left (\mathit {a2} +2\right )}{2}\right )}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right )+\mathit {a1}^{2} \mathit {a3}^{2} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} -\mathit {a1} \mathit {a2} \right ) \KummerM \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a0} \mathit {a3} \mathit {a1} \sqrt {-\mathit {a1} \mathit {a3}}+\mathit {a1}^{2} \mathit {a2} \mathit {a3}}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right )-\frac {\KummerU \left (\frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +2 \mathit {a0} \mathit {a3} \mathit {a1} \sqrt {-\mathit {a1} \mathit {a3}}+\mathit {a1}^{2} \mathit {a2} \mathit {a3}}{2 \mathit {a1}^{2} \mathit {a3}}, \frac {\left (-\mathit {a1} \mathit {a3} \right )^{\frac {3}{2}} \mathit {a0} +\mathit {a1} \mathit {a3} \left (\sqrt {-\mathit {a1} \mathit {a3}}\, \mathit {a0} +\mathit {a1} \left (1+\mathit {a2} \right )\right )}{\mathit {a1}^{2} \mathit {a3}}, 2 x \sqrt {-\mathit {a1} \mathit {a3}}\right ) \textit {\_C1} \left (\mathit {a2} \sqrt {-\mathit {a1} \mathit {a3}}+\mathit {a3} \mathit {a0} \right )}{2}\right ) \mathit {a1}^{4}}\right ]$ Mathematica raw input

DSolve[a0 + a1*x + y[x]*(a2 + a3*x*y[x]) + x*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> ((-I)*(Sqrt[a1]*C[1]*HypergeometricU[(a2 + (I*a0*Sqrt[a3])/Sqrt[a1])/2
, a2, (2*I)*Sqrt[a1]*Sqrt[a3]*x] + (Sqrt[a1]*a2 + I*a0*Sqrt[a3])*C[1]*Hypergeome
tricU[(2 + a2 + (I*a0*Sqrt[a3])/Sqrt[a1])/2, 1 + a2, (2*I)*Sqrt[a1]*Sqrt[a3]*x]
+ Sqrt[a1]*(2*LaguerreL[-1 - a2/2 - ((I/2)*a0*Sqrt[a3])/Sqrt[a1], a2, (2*I)*Sqrt
[a1]*Sqrt[a3]*x] + LaguerreL[-1/2*a2 - ((I/2)*a0*Sqrt[a3])/Sqrt[a1], -1 + a2, (2
*I)*Sqrt[a1]*Sqrt[a3]*x])))/(Sqrt[a3]*(C[1]*HypergeometricU[(a2 + (I*a0*Sqrt[a3]
)/Sqrt[a1])/2, a2, (2*I)*Sqrt[a1]*Sqrt[a3]*x] + LaguerreL[-1/2*a2 - ((I/2)*a0*Sq
rt[a3])/Sqrt[a1], -1 + a2, (2*I)*Sqrt[a1]*Sqrt[a3]*x]))}}

Maple raw input

dsolve(x*diff(y(x),x)+a0+a1*x+(a2+a3*x*y(x))*y(x) = 0, y(x))

Maple raw output

[y(x) = 4*a1^2*(-1/4*_C1*(a0^2*a1^2*a3^3+a1^3*a2^2*a3^2-2*(-a1*a3)^(3/2)*a0*a1*a
2*a3-2*(-a1*a3)^(5/2)*a0*a2)*KummerU(1/2*((-a1*a3)^(3/2)*a0+2*a1*a3*((-a1*a3)^(1
/2)*a0+1/2*a1*(a2+2)))/a1^2/a3,((-a1*a3)^(3/2)*a0+a1*a3*((-a1*a3)^(1/2)*a0+a1*(1
+a2)))/a1^2/a3,2*x*(-a1*a3)^(1/2))+(a1^3*a3*(a3*a0-a2*(-a1*a3)^(1/2))*KummerM(1/
2*((-a1*a3)^(3/2)*a0+2*a1*a3*((-a1*a3)^(1/2)*a0+1/2*a1*(a2+2)))/a1^2/a3,((-a1*a3
)^(3/2)*a0+a1*a3*((-a1*a3)^(1/2)*a0+a1*(1+a2)))/a1^2/a3,2*x*(-a1*a3)^(1/2))+a1^3
*a3*(a2*(-a1*a3)^(1/2)+a3*a0)*KummerM(1/2/a1^2/a3*((-a1*a3)^(3/2)*a0+2*a0*a3*a1*
(-a1*a3)^(1/2)+a1^2*a2*a3),((-a1*a3)^(3/2)*a0+a1*a3*((-a1*a3)^(1/2)*a0+a1*(1+a2)
))/a1^2/a3,2*x*(-a1*a3)^(1/2))+1/2*KummerU(1/2/a1^2/a3*((-a1*a3)^(3/2)*a0+2*a0*a
3*a1*(-a1*a3)^(1/2)+a1^2*a2*a3),((-a1*a3)^(3/2)*a0+a1*a3*((-a1*a3)^(1/2)*a0+a1*(
1+a2)))/a1^2/a3,2*x*(-a1*a3)^(1/2))*_C1*((-a1*a3)^(1/2)*a0-a1*a2))*a3^2*a1^2)/(_
C1*(a1^4*a2^2*a3^2*(-a1*a3)^(1/2)+2*a0^2*a1*a3*(-a1*a3)^(5/2)+(-a1*a3)^(7/2)*a0^
2)*KummerU(1/2*((-a1*a3)^(3/2)*a0+2*a1*a3*((-a1*a3)^(1/2)*a0+1/2*a1*(a2+2)))/a1^
2/a3,((-a1*a3)^(3/2)*a0+a1*a3*((-a1*a3)^(1/2)*a0+a1*(1+a2)))/a1^2/a3,2*x*(-a1*a3
)^(1/2))+4*a3^2*(-a1^2*a3^2*((-a1*a3)^(1/2)*a0+a1*a2)*KummerM(1/2*((-a1*a3)^(3/2
)*a0+2*a1*a3*((-a1*a3)^(1/2)*a0+1/2*a1*(a2+2)))/a1^2/a3,((-a1*a3)^(3/2)*a0+a1*a3
*((-a1*a3)^(1/2)*a0+a1*(1+a2)))/a1^2/a3,2*x*(-a1*a3)^(1/2))+a1^2*a3^2*((-a1*a3)^
(1/2)*a0-a1*a2)*KummerM(1/2/a1^2/a3*((-a1*a3)^(3/2)*a0+2*a0*a3*a1*(-a1*a3)^(1/2)
+a1^2*a2*a3),((-a1*a3)^(3/2)*a0+a1*a3*((-a1*a3)^(1/2)*a0+a1*(1+a2)))/a1^2/a3,2*x
*(-a1*a3)^(1/2))-1/2*KummerU(1/2/a1^2/a3*((-a1*a3)^(3/2)*a0+2*a0*a3*a1*(-a1*a3)^
(1/2)+a1^2*a2*a3),((-a1*a3)^(3/2)*a0+a1*a3*((-a1*a3)^(1/2)*a0+a1*(1+a2)))/a1^2/a
3,2*x*(-a1*a3)^(1/2))*_C1*(a2*(-a1*a3)^(1/2)+a3*a0))*a1^4)]