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 = 0.796925 (sec), leaf count = 744
\[\left \{\left \{y(x)\to \frac {2^{2/3} \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\frac {2 \sqrt [3]{2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-2 b x}{2 c}\right \},\left \{y(x)\to \frac {9 i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\frac {18 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^2 \left (a c-b^2\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-36 b x}{36 c}\right \},\left \{y(x)\to \frac {-9\ 2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}+\frac {18 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (3 a b c x^3-2 b^3 x^3+c^2 \left (-k x^3+e^{3 c_1}\right )\right ){}^2}+3 a b c x^3-2 b^3 x^3-c^2 k x^3+c^2 e^{3 c_1}}}-36 b x}{36 c}\right \}\right \}\]
Maple ✓
cpu = 0.257 (sec), leaf count = 1666
\[\left [y \left (x \right ) = \frac {\frac {\left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{2 c}-\frac {2 \textit {\_C1}^{2} x^{2} \left (c a -b^{2}\right )}{c \left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}-\frac {\textit {\_C1} b x}{c}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{4 c}+\frac {\textit {\_C1}^{2} x^{2} \left (c a -b^{2}\right )}{c \left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}-\frac {\textit {\_C1} b x}{c}-\frac {i \sqrt {3}\, \left (\frac {\left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{2 c}+\frac {2 \textit {\_C1}^{2} x^{2} \left (c a -b^{2}\right )}{c \left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}, y \left (x \right ) = \frac {-\frac {\left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{4 c}+\frac {\textit {\_C1}^{2} x^{2} \left (c a -b^{2}\right )}{c \left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}-\frac {\textit {\_C1} b x}{c}+\frac {i \sqrt {3}\, \left (\frac {\left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}{2 c}+\frac {2 \textit {\_C1}^{2} x^{2} \left (c a -b^{2}\right )}{c \left (12 \textit {\_C1}^{3} a b c \,x^{3}-8 \textit {\_C1}^{3} b^{3} x^{3}-4 \textit {\_C1}^{3} c^{2} k \,x^{3}+4 \sqrt {4 \textit {\_C1}^{6} a^{3} c \,x^{6}-3 \textit {\_C1}^{6} a^{2} b^{2} x^{6}-6 \textit {\_C1}^{6} a b c k \,x^{6}+4 \textit {\_C1}^{6} b^{3} k \,x^{6}+\textit {\_C1}^{6} c^{2} k^{2} x^{6}+6 \textit {\_C1}^{3} a b c \,x^{3}-4 \textit {\_C1}^{3} b^{3} x^{3}-2 \textit {\_C1}^{3} c^{2} k \,x^{3}+c^{2}}\, c +4 c^{2}\right )^{\frac {1}{3}}}\right )}{2}}{\textit {\_C1}}\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))
Maple raw output
[y(x) = (1/2/c*(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)-2*_C1^2*x^2*(a*c-b^2)/c/(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)-_C1*b*x/c)/_C1, y(x) = (-1/4/c*(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)+_C1^2*x^2*(a*c-b^2)/c/(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)-_C1*b*x/c-1/2*I*3^(1/2)*(1/2/c*(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)+2*_C1^2*x^2*(a*c-b^2)/c/(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)))/_C1, y(x) = (-1/4/c*(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)+_C1^2*x^2*(a*c-b^2)/c/(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)-_C1*b*x/c+1/2*I*3^(1/2)*(1/2/c*(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)+2*_C1^2*x^2*(a*c-b^2)/c/(12*_C1^3*a*b*c*x^3-8*_C1^3*b^3*x^3-4*_C1^3*c^2*k*x^3+4*(4*_C1^6*a^3*c*x^6-3*_C1^6*a^2*b^2*x^6-6*_C1^6*a*b*c*k*x^6+4*_C1^6*b^3*k*x^6+_C1^6*c^2*k^2*x^6+6*_C1^3*a*b*c*x^3-4*_C1^3*b^3*x^3-2*_C1^3*c^2*k*x^3+c^2)^(1/2)*c+4*c^2)^(1/3)))/_C1]