Internal problem ID [7870]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 290.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _exact, _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (a y^{2}+2 b x y+c \,x^{2}\right ) y^{\prime }+b y^{2}+2 c x y+d \,x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.066 (sec). Leaf size: 1666
dsolve((a*y(x)^2+2*b*x*y(x)+c*x^2)*diff(y(x),x)+b*y(x)^2+2*c*x*y(x)+d*x^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {2 c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{a \left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}-\frac {b x c_{1}}{a}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}{4 a}+\frac {c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{a \left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}-\frac {b x c_{1}}{a}-\frac {i \sqrt {3}\, \left (\frac {\left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {2 c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{a \left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ y \relax (x ) = \frac {-\frac {\left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}{4 a}+\frac {c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{a \left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}-\frac {b x c_{1}}{a}+\frac {i \sqrt {3}\, \left (\frac {\left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {2 c_{1}^{2} x^{2} \left (a c -b^{2}\right )}{a \left (-4 c_{1}^{3} a^{2} d \,x^{3}+12 c \,x^{3} c_{1}^{3} b a -8 b^{3} x^{3} c_{1}^{3}+4 \sqrt {a^{2} c_{1}^{6} d^{2} x^{6}-6 a b c c_{1}^{6} d \,x^{6}+4 a \,c^{3} c_{1}^{6} x^{6}+4 b^{3} c_{1}^{6} d \,x^{6}-3 b^{2} c^{2} c_{1}^{6} x^{6}-2 c_{1}^{3} a^{2} d \,x^{3}+6 c \,x^{3} c_{1}^{3} b a -4 b^{3} x^{3} c_{1}^{3}+a^{2}}\, a +4 a^{2}\right )^{\frac {1}{3}}}\right )}{2}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.032 (sec). Leaf size: 1327
DSolve[(a*y[x]^2+2*b*x*y[x]+c*x^2)*y'[x]+b*y[x]^2+2*c*x*y[x]+d*x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{2/3} \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (a^2 d-3 a b c+2 b^3\right )+a^2 e^{3 c_1}\right ){}^2}+a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3}+\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 (-x^3 \left (a^2 d-3 a b c+2 b^3\right )+a^2 e^{3 c_1}\right ){}^2}+a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3}}-2 b x}{2 a} \\ 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 (-x^3 \left (a^2 d-3 a b c+2 b^3\right )+a^2 e^{3 c_1}\right ){}^2}+a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3}+\frac {36 \sqrt [3]{-2} x^2 \left (a c-b^2\right )}{\sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (a^2 d-3 a b c+2 b^3\right )+a^2 e^{3 c_1}\right ){}^2}+a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3}}-36 b x}{36 a} \\ y(x)\to \frac {-9\operatorname {\ }2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {-4 x^6 \left (b^2-a c\right )^3+\left (-x^3 \left (a^2 d-3 a b c+2 b^3\right )+a^2 e^{3 c_1}\right ){}^2}+a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3}+\frac {36 (-1)^{2/3} \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 (-x^3 \left (a^2 d-3 a b c+2 b^3\right )+a^2 e^{3 c_1}\right ){}^2}+a^2 \left (-d x^3+e^{3 c_1}\right )+3 a b c x^3-2 b^3 x^3}}-36 b x}{36 a} \\ y(x)\to \frac {-2 b x \sqrt [3]{\sqrt {a^2 x^6 \left (a^2 d^2-6 a b c d+4 a c^3+4 b^3 d-3 b^2 c^2\right )}-x^3 \left (a^2 d-3 a b c+2 b^3\right )}+\left (2 \sqrt {a^2 x^6 \left (a^2 d^2-6 a b c d+4 a c^3+4 b^3 d-3 b^2 c^2\right )}-2 x^3 \left (a^2 d-3 a b c+2 b^3\right )\right )^{2/3}+2 \sqrt [3]{2} x^2 \left (b^2-a c\right )}{2 a \sqrt [3]{\sqrt {a^2 x^6 \left (a^2 d^2-6 a b c d+4 a c^3+4 b^3 d-3 b^2 c^2\right )}-x^3 \left (a^2 d-3 a b c+2 b^3\right )}} \\ y(x)\to \frac {-2 \sqrt [3]{-1} 2^{2/3} \sqrt [3]{\sqrt {a^2 x^6 \left (a^2 d^2-6 a b c d+4 a c^3+4 b^3 d-3 b^2 c^2\right )}-x^3 \left (a^2 d-3 a b c+2 b^3\right )}+\frac {4 (-1)^{2/3} \sqrt [3]{2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {a^2 x^6 \left (a^2 d^2-6 a b c d+4 a c^3+4 b^3 d-3 b^2 c^2\right )}-x^3 \left (a^2 d-3 a b c+2 b^3\right )}}-4 b x}{4 a} \\ y(x)\to \frac {(-2)^{2/3} \sqrt [3]{\sqrt {a^2 x^6 \left (a^2 d^2-6 a b c d+4 a c^3+4 b^3 d-3 b^2 c^2\right )}-x^3 \left (a^2 d-3 a b c+2 b^3\right )}-\frac {2 \sqrt [3]{-2} x^2 \left (b^2-a c\right )}{\sqrt [3]{\sqrt {a^2 x^6 \left (a^2 d^2-6 a b c d+4 a c^3+4 b^3 d-3 b^2 c^2\right )}-x^3 \left (a^2 d-3 a b c+2 b^3\right )}}-2 b x}{2 a} \\ \end{align*}