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

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

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

Book solution method
Exact equation

Mathematica
cpu = 0.0705899 (sec), leaf count = 744

\[\left \{\left \{y(x)\to \frac {2^{2/3} \sqrt [3]{\sqrt {\left (3 a b c x^3-2 b^3 x^3+c^2 \left (e^{3 c_1}-k x^3\right )\right ){}^2-4 x^6 \left (b^2-a c\right )^3}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\frac {2 \sqrt [3]{2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {\left (3 a b c x^3-2 b^3 x^3+c^2 \left (e^{3 c_1}-k x^3\right )\right ){}^2-4 x^6 \left (b^2-a c\right )^3}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-2 b x}{2 c}\right \},\left \{y(x)\to \frac {9 i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {\left (3 a b c x^3-2 b^3 x^3+c^2 \left (e^{3 c_1}-k x^3\right )\right ){}^2-4 x^6 \left (b^2-a c\right )^3}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\frac {18 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2 \left (a c-b^2\right )}{\sqrt [3]{\sqrt {\left (3 a b c x^3-2 b^3 x^3+c^2 \left (e^{3 c_1}-k x^3\right )\right ){}^2-4 x^6 \left (b^2-a c\right )^3}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-36 b x}{36 c}\right \},\left \{y(x)\to \frac {-9\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (3 a b c x^3-2 b^3 x^3+c^2 \left (e^{3 c_1}-k x^3\right )\right ){}^2-4 x^6 \left (b^2-a c\right )^3}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\frac {18 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {\left (3 a b c x^3-2 b^3 x^3+c^2 \left (e^{3 c_1}-k x^3\right )\right ){}^2-4 x^6 \left (b^2-a c\right )^3}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-36 b x}{36 c}\right \}\right \}\]

Maple
cpu = 0.015 (sec), leaf count = 46

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

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

Mathematica raw output

{{y[x] -> (-2*b*x + (2*2^(1/3)*(b^2 - a*c)*x^2)/(c^2*E^(3*C[1]) - 2*b^3*x^3 + 3*
a*b*c*x^3 - c^2*k*x^3 + Sqrt[-4*(b^2 - a*c)^3*x^6 + (-2*b^3*x^3 + 3*a*b*c*x^3 + 
c^2*(E^(3*C[1]) - k*x^3))^2])^(1/3) + 2^(2/3)*(c^2*E^(3*C[1]) - 2*b^3*x^3 + 3*a*
b*c*x^3 - c^2*k*x^3 + Sqrt[-4*(b^2 - a*c)^3*x^6 + (-2*b^3*x^3 + 3*a*b*c*x^3 + c^
2*(E^(3*C[1]) - k*x^3))^2])^(1/3))/(2*c)}, {y[x] -> (-36*b*x + (18*2^(1/3)*(1 + 
I*Sqrt[3])*(-b^2 + a*c)*x^2)/(c^2*E^(3*C[1]) - 2*b^3*x^3 + 3*a*b*c*x^3 - c^2*k*x
^3 + Sqrt[-4*(b^2 - a*c)^3*x^6 + (-2*b^3*x^3 + 3*a*b*c*x^3 + c^2*(E^(3*C[1]) - k
*x^3))^2])^(1/3) + (9*I)*2^(2/3)*(I + Sqrt[3])*(c^2*E^(3*C[1]) - 2*b^3*x^3 + 3*a
*b*c*x^3 - c^2*k*x^3 + Sqrt[-4*(b^2 - a*c)^3*x^6 + (-2*b^3*x^3 + 3*a*b*c*x^3 + c
^2*(E^(3*C[1]) - k*x^3))^2])^(1/3))/(36*c)}, {y[x] -> (-36*b*x + ((18*I)*2^(1/3)
*(I + Sqrt[3])*(b^2 - a*c)*x^2)/(c^2*E^(3*C[1]) - 2*b^3*x^3 + 3*a*b*c*x^3 - c^2*
k*x^3 + Sqrt[-4*(b^2 - a*c)^3*x^6 + (-2*b^3*x^3 + 3*a*b*c*x^3 + c^2*(E^(3*C[1]) 
- k*x^3))^2])^(1/3) - 9*2^(2/3)*(1 + I*Sqrt[3])*(c^2*E^(3*C[1]) - 2*b^3*x^3 + 3*
a*b*c*x^3 - c^2*k*x^3 + Sqrt[-4*(b^2 - a*c)^3*x^6 + (-2*b^3*x^3 + 3*a*b*c*x^3 + 
c^2*(E^(3*C[1]) - k*x^3))^2])^(1/3))/(36*c)}}

Maple raw input

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

Maple raw output

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