Internal problem ID [7893]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 313.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\left (2 a y^{3}+3 a x y^{2}-b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 b \,x^{2} y+2 b \,x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 912
dsolve((2*a*y(x)^3+3*a*x*y(x)^2-b*x^3+c*x^2)*diff(y(x),x)-a*y(x)^3+c*y(x)^2+3*b*x^2*y(x)+2*b*x^3 = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}{6 a}+\frac {-2 c x +2 c_{1}}{\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}{12 a}-\frac {-c x +c_{1}}{\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}{6 a}-\frac {2 \left (-c x +c_{1}\right )}{\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}{12 a}-\frac {-c x +c_{1}}{\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}{6 a}-\frac {2 \left (-c x +c_{1}\right )}{\left (\left (-108 b \,x^{3}+108 c_{1} x +12 \sqrt {-\frac {3 \left (-27 a \,b^{2} x^{6}+54 a b \,x^{4} c_{1}-4 c^{3} x^{3}-27 a \,x^{2} c_{1}^{2}+12 c^{2} x^{2} c_{1}-12 c x c_{1}^{2}+4 c_{1}^{3}\right )}{a}}\right ) a^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.388 (sec). Leaf size: 495
DSolve[2*b*x^3 + 3*b*x^2*y[x] + c*y[x]^2 - a*y[x]^3 + (c*x^2 - b*x^3 + 3*a*x*y[x]^2 + 2*a*y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2 \sqrt [3]{3} a (c x+c_1)-\sqrt [3]{2} \left (\sqrt {3} \sqrt {a^3 \left (27 a x^2 \left (b x^2+c_1\right ){}^2+4 (c x+c_1){}^3\right )}+9 a^2 x \left (b x^2+c_1\right )\right ){}^{2/3}}{6^{2/3} a \sqrt [3]{\sqrt {3} \sqrt {a^3 \left (27 a x^2 \left (b x^2+c_1\right ){}^2+4 (c x+c_1){}^3\right )}+9 a^2 x \left (b x^2+c_1\right )}} \\ y(x)\to \frac {\left (-1-i \sqrt {3}\right ) (c x+c_1)}{2^{2/3} \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 x \left (b x^2+c_1\right )}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 x \left (b x^2+c_1\right )}}{6 \sqrt [3]{2} a} \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) (c x+c_1)}{2^{2/3} \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 x \left (b x^2+c_1\right )}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {\left (27 a^2 b x^3+27 a^2 c_1 x\right ){}^2+4 (3 a c x+3 a c_1){}^3}+27 a^2 x \left (b x^2+c_1\right )}}{6 \sqrt [3]{2} a} \\ \end{align*}