4.13.39 \(y'(x) \left (x^2 (a+2 b)+2 x (2 b+c) y(x)+3 c y(x)^2\right )+2 x (a+2 b) y(x)+3 a x^2+(2 b+c) y(x)^2=0\)

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

[[_homogeneous, `class A`], _exact, _rational, _dAlembert]

Book solution method
Exact equation

Mathematica
cpu = 0.107888 (sec), leaf count = 1105

\[\left \{\left \{y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (4 b^2-2 c b+c (c-3 a)\right ) x^2}{\sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}-2 (2 b+c) x+2^{2/3} \sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}{6 c}\right \},\left \{y(x)\to \frac {-\frac {2 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) \left (4 b^2-2 c b+c (c-3 a)\right ) x^2}{\sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}-4 (2 b+c) x+i 2^{2/3} \left (i+\sqrt {3}\right ) \sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}{12 c}\right \},\left \{y(x)\to -\frac {\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) \left (4 b^2-2 c b+c (c-3 a)\right ) x^2}{\sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}+4 (2 b+c) x+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-16 b^3 x^3-2 c^3 x^3-18 a c^2 x^3+6 b c^2 x^3+12 b^2 c x^3+18 a b c x^3+27 c^2 e^{c_1}+\sqrt {4 \left (3 (a+2 b) c-(2 b+c)^2\right )^3 x^6+\left (16 b^3 x^3+2 c^3 x^3-6 b (3 a+2 b) c x^3-3 c^2 \left (2 (b-3 a) x^3+9 e^{c_1}\right )\right ){}^2}}}{12 c}\right \}\right \}\]

Maple
cpu = 0.017 (sec), leaf count = 40

\[ \left \{ -{\frac {1}{3}\ln \left ({\frac { \left (x+y \relax (x ) \right ) \left (a{x}^{2}+2\,bxy \relax (x ) +c \left (y \relax (x ) \right ) ^{2} \right ) }{{x}^{3}}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-2*(2*b + c)*x + (2*2^(1/3)*(4*b^2 - 2*b*c + c*(-3*a + c))*x^2)/(27*c
^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^
3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(
3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3))^2])^(1/3) + 
2^(2/3)*(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x^3
 + 6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*b^
3*x^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3))
^2])^(1/3))/(6*c)}, {y[x] -> (-4*(2*b + c)*x - ((2*I)*2^(1/3)*(-I + Sqrt[3])*(4*
b^2 - 2*b*c + c*(-3*a + c))*x^2)/(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12
*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2
*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^
C[1] + 2*(-3*a + b)*x^3))^2])^(1/3) + I*2^(2/3)*(I + Sqrt[3])*(27*c^2*E^C[1] - 1
6*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^3 - 2*c^3*x^3
 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(3*a + 2*b)*c*
x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3))^2])^(1/3))/(12*c)}, {y[x]
 -> -(4*(2*b + c)*x + (2*2^(1/3)*(1 - I*Sqrt[3])*(4*b^2 - 2*b*c + c*(-3*a + c))*
x^2)/(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 12*b^2*c*x^3 - 18*a*c^2*x^3 + 
6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (2*b + c)^2)^3*x^6 + (16*b^3*x
^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E^C[1] + 2*(-3*a + b)*x^3))^2]
)^(1/3) + 2^(2/3)*(1 + I*Sqrt[3])*(27*c^2*E^C[1] - 16*b^3*x^3 + 18*a*b*c*x^3 + 1
2*b^2*c*x^3 - 18*a*c^2*x^3 + 6*b*c^2*x^3 - 2*c^3*x^3 + Sqrt[4*(3*(a + 2*b)*c - (
2*b + c)^2)^3*x^6 + (16*b^3*x^3 - 6*b*(3*a + 2*b)*c*x^3 + 2*c^3*x^3 - 3*c^2*(9*E
^C[1] + 2*(-3*a + b)*x^3))^2])^(1/3))/(12*c)}}

Maple raw input

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

Maple raw output

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