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 = 0.180795 (sec), leaf count = 301

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

Maple
cpu = 0.336 (sec), leaf count = 848

\[ \left \{ y \relax (x ) =4\,{{{\it a1}}^{2} \left (-1/4\,{\it \_C1}\, \left ({{\it a0}}^{2}{{\it a1}}^{2}{{\it a3}}^{3}+{{\it a1}}^{3}{{\it a2}}^{2}{{\it a3}}^{2}-2\, \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}\,{\it a1}\,{\it a2}\,{\it a3}-2\, \left (-{\it a1}\,{\it a3} \right ) ^{5/2}{\it a0}\,{\it a2} \right ) {{\sl U}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+1/2\,{\it a1}\, \left ({\it a2}+2 \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}+{{\it a3}}^{2} \left ({{\it a1}}^{3}{\it a3}\, \left ({\it a3}\,{\it a0}-{\it a2}\,\sqrt {-{\it a1}\,{\it a3}} \right ) {{\sl M}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+1/2\,{\it a1}\, \left ({\it a2}+2 \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}+{{\it a1}}^{3}{\it a3}\, \left ({\it a2}\,\sqrt {-{\it a1}\,{\it a3}}+{\it a3}\,{\it a0} \right ) {{\sl M}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a0}\,{\it a3}\,{\it a1}\,\sqrt {-{\it a1}\,{\it a3}}+{{\it a1}}^{2}{\it a2}\,{\it a3}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}+1/2\,{{\sl U}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a0}\,{\it a3}\,{\it a1}\,\sqrt {-{\it a1}\,{\it a3}}+{{\it a1}}^{2}{\it a2}\,{\it a3}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}{\it \_C1}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}-{\it a1}\,{\it a2} \right ) \right ) {{\it a1}}^{2} \right ) \left ({\it \_C1}\, \left ({{\it a1}}^{4}{{\it a2}}^{2}{{\it a3}}^{2}\sqrt {-{\it a1}\,{\it a3}}+2\,{{\it a0}}^{2}{\it a1}\,{\it a3}\, \left (-{\it a1}\,{\it a3} \right ) ^{5/2}+ \left (-{\it a1}\,{\it a3} \right ) ^{7/2}{{\it a0}}^{2} \right ) {{\sl U}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+1/2\,{\it a1}\, \left ({\it a2}+2 \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}+4\, \left (-{{\it a1}}^{2}{{\it a3}}^{2} \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\,{\it a2} \right ) {{\sl M}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+1/2\,{\it a1}\, \left ({\it a2}+2 \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}+{{\it a1}}^{2}{{\it a3}}^{2} \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}-{\it a1}\,{\it a2} \right ) {{\sl M}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a0}\,{\it a3}\,{\it a1}\,\sqrt {-{\it a1}\,{\it a3}}+{{\it a1}}^{2}{\it a2}\,{\it a3}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}-1/2\,{{\sl U}\left (1/2\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+2\,{\it a0}\,{\it a3}\,{\it a1}\,\sqrt {-{\it a1}\,{\it a3}}+{{\it a1}}^{2}{\it a2}\,{\it a3}}{{{\it a1}}^{2}{\it a3}}},\,{\frac { \left (-{\it a1}\,{\it a3} \right ) ^{3/2}{\it a0}+{\it a3}\, \left (\sqrt {-{\it a1}\,{\it a3}}{\it a0}+{\it a1}\, \left (1+{\it a2} \right ) \right ) {\it a1}}{{{\it a1}}^{2}{\it a3}}},\,2\,x\sqrt {-{\it a1}\,{\it a3}}\right )}{\it \_C1}\, \left ({\it a2}\,\sqrt {-{\it a1}\,{\it a3}}+{\it a3}\,{\it a0} \right ) \right ) {{\it a3}}^{2}{{\it a1}}^{4} \right ) ^{-1}} \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[-a2/2 - ((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[-a2/2 - ((I/2)*a0*Sqrt[a
3])/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),'implicit')

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*a2
*a3-2*(-a1*a3)^(5/2)*a0*a2)*KummerU(1/2*((-a1*a3)^(3/2)*a0+2*a3*((-a1*a3)^(1/2)*
a0+1/2*a1*(a2+2))*a1)/a1^2/a3,((-a1*a3)^(3/2)*a0+a3*((-a1*a3)^(1/2)*a0+a1*(1+a2)
)*a1)/a1^2/a3,2*x*(-a1*a3)^(1/2))+a3^2*(a1^3*a3*(a3*a0-a2*(-a1*a3)^(1/2))*Kummer
M(1/2*((-a1*a3)^(3/2)*a0+2*a3*((-a1*a3)^(1/2)*a0+1/2*a1*(a2+2))*a1)/a1^2/a3,((-a
1*a3)^(3/2)*a0+a3*((-a1*a3)^(1/2)*a0+a1*(1+a2))*a1)/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+a3*((-a1*a3)^(1/2)*a0+a1*(1+a2
))*a1)/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+a3*((-a1*a3)^(1/2)*a0+a1*
(1+a2))*a1)/a1^2/a3,2*x*(-a1*a3)^(1/2))*_C1*((-a1*a3)^(1/2)*a0-a1*a2))*a1^2)/(_C
1*(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*a3*((-a1*a3)^(1/2)*a0+1/2*a1*(a2+2))*a1)/a1^2
/a3,((-a1*a3)^(3/2)*a0+a3*((-a1*a3)^(1/2)*a0+a1*(1+a2))*a1)/a1^2/a3,2*x*(-a1*a3)
^(1/2))+4*(-a1^2*a3^2*((-a1*a3)^(1/2)*a0+a1*a2)*KummerM(1/2*((-a1*a3)^(3/2)*a0+2
*a3*((-a1*a3)^(1/2)*a0+1/2*a1*(a2+2))*a1)/a1^2/a3,((-a1*a3)^(3/2)*a0+a3*((-a1*a3
)^(1/2)*a0+a1*(1+a2))*a1)/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+a3*((-a1*a3)^(1/2)*a0+a1*(1+a2))*a1)/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+a3*((-a1*a3)^(1/2)*a0+a1*(1+a2))*a1)/a1^2/a3,2*x*
(-a1*a3)^(1/2))*_C1*(a2*(-a1*a3)^(1/2)+a3*a0))*a3^2*a1^4)