4.12.39 \(x \left (2 x^2 y(x)+3\right ) y'(x)+y(x) \left (3 x^2 y(x)+4\right )=0\)

ODE
\[ x \left (2 x^2 y(x)+3\right ) y'(x)+y(x) \left (3 x^2 y(x)+4\right )=0 \] ODE Classification

[[_homogeneous, `class G`], _rational, [_Abel, `2nd type``class B`]]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 1.41743 (sec), leaf count = 1769

\[\left \{\left \{y(x)\to \frac {\sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}{2 \sqrt {3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}-\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}+\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}-\frac {1}{2 x^2}\right \},\left \{y(x)\to \frac {\sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}{2 \sqrt {3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}-\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}+\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}-\frac {1}{2 x^2}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}{2 \sqrt {3}}-\frac {1}{2} \sqrt {-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}+\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}+\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}-\frac {1}{2 x^2}\right \},\left \{y(x)\to -\frac {\sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}{2 \sqrt {3}}+\frac {1}{2} \sqrt {-\frac {\sqrt [3]{2} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{3^{2/3} x^6}+\frac {2 \sqrt {3}}{x^6 \sqrt {\frac {\sqrt [3]{6} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}{x^6}+\frac {3}{x^4}-\frac {2\ 6^{2/3} e^{-2 c_1}}{\sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}}+\frac {2}{x^4}+\frac {2\ 2^{2/3} e^{-2 c_1}}{\sqrt [3]{3} \sqrt [3]{e^{-6 c_1} \left (\sqrt {48 e^{6 c_1} x^{18}+81 e^{8 c_1} x^{16}}-9 e^{4 c_1} x^8\right )}}}-\frac {1}{2 x^2}\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 29

\[ \left \{ \ln \relax (x ) -{\it \_C1}-{\frac {\ln \left ({x}^{2}y \relax (x ) +2 \right ) }{2}}-{\frac {3\,\ln \left ({x}^{2}y \relax (x ) \right ) }{2}}=0 \right \} \] Mathematica raw input

DSolve[y[x]*(4 + 3*x^2*y[x]) + x*(3 + 2*x^2*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -1/(2*x^2) + Sqrt[3/x^4 - (2*6^(2/3))/(E^(2*C[1])*((-9*E^(4*C[1])*x^8 
+ Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) + (6^(1/3)*(
(-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^
(1/3))/x^6]/(2*Sqrt[3]) - Sqrt[2/x^4 + (2*2^(2/3))/(3^(1/3)*E^(2*C[1])*((-9*E^(4
*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) -
 (2^(1/3)*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E
^(6*C[1]))^(1/3))/(3^(2/3)*x^6) - (2*Sqrt[3])/(x^6*Sqrt[3/x^4 - (2*6^(2/3))/(E^(
2*C[1])*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(
6*C[1]))^(1/3)) + (6^(1/3)*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^
(6*C[1])*x^18])/E^(6*C[1]))^(1/3))/x^6])]/2}, {y[x] -> -1/(2*x^2) + Sqrt[3/x^4 -
 (2*6^(2/3))/(E^(2*C[1])*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6
*C[1])*x^18])/E^(6*C[1]))^(1/3)) + (6^(1/3)*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C
[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3))/x^6]/(2*Sqrt[3]) + Sqrt[2/x^
4 + (2*2^(2/3))/(3^(1/3)*E^(2*C[1])*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^1
6 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) - (2^(1/3)*((-9*E^(4*C[1])*x^8 + Sqr
t[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3))/(3^(2/3)*x^6) - (
2*Sqrt[3])/(x^6*Sqrt[3/x^4 - (2*6^(2/3))/(E^(2*C[1])*((-9*E^(4*C[1])*x^8 + Sqrt[
81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) + (6^(1/3)*((-9*E^(
4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3))/
x^6])]/2}, {y[x] -> -1/(2*x^2) - Sqrt[3/x^4 - (2*6^(2/3))/(E^(2*C[1])*((-9*E^(4*
C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) + 
(6^(1/3)*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^
(6*C[1]))^(1/3))/x^6]/(2*Sqrt[3]) - Sqrt[2/x^4 + (2*2^(2/3))/(3^(1/3)*E^(2*C[1])
*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1])
)^(1/3)) - (2^(1/3)*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1]
)*x^18])/E^(6*C[1]))^(1/3))/(3^(2/3)*x^6) + (2*Sqrt[3])/(x^6*Sqrt[3/x^4 - (2*6^(
2/3))/(E^(2*C[1])*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*
x^18])/E^(6*C[1]))^(1/3)) + (6^(1/3)*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^
16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3))/x^6])]/2}, {y[x] -> -1/(2*x^2) - Sq
rt[3/x^4 - (2*6^(2/3))/(E^(2*C[1])*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16
 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) + (6^(1/3)*((-9*E^(4*C[1])*x^8 + Sqrt
[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3))/x^6]/(2*Sqrt[3]) +
 Sqrt[2/x^4 + (2*2^(2/3))/(3^(1/3)*E^(2*C[1])*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8
*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) - (2^(1/3)*((-9*E^(4*C[1])
*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3))/(3^(2/3
)*x^6) + (2*Sqrt[3])/(x^6*Sqrt[3/x^4 - (2*6^(2/3))/(E^(2*C[1])*((-9*E^(4*C[1])*x
^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]))^(1/3)) + (6^(1/3
)*((-9*E^(4*C[1])*x^8 + Sqrt[81*E^(8*C[1])*x^16 + 48*E^(6*C[1])*x^18])/E^(6*C[1]
))^(1/3))/x^6])]/2}}

Maple raw input

dsolve(x*(3+2*x^2*y(x))*diff(y(x),x)+(4+3*x^2*y(x))*y(x) = 0, y(x),'implicit')

Maple raw output

ln(x)-_C1-1/2*ln(x^2*y(x)+2)-3/2*ln(x^2*y(x)) = 0