Internal problem ID [8520]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 184.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (a \,x^{2}+x b +c \right )^{2} \left (y^{\prime }+y^{2}\right )=-A} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 846
dsolve((a*x^2+b*x+c)^2*(diff(y(x),x)+y(x)^2) + A=0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2 \left (-i \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}\, \sqrt {4 a c -b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} a +i \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}\, \sqrt {4 a c -b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} a +2 \sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} a x +2 \sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} a x +\sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} b +\sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} b \right ) a}{\sqrt {-4 a c +b^{2}}\, \left (2 a x +b +i \sqrt {4 a c -b^{2}}\right ) \left (i \sqrt {4 a c -b^{2}}-2 a x -b \right ) \left (c_{1} {\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {-\frac {4 a c -b^{2}+4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )} \]
✓ Solution by Mathematica
Time used: 3.439 (sec). Leaf size: 743
DSolve[(a*x^2+b*x+c)^2*(y'[x]+y[x]^2) + A==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b^2 c_1 \left (-\exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )+b c_1 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+4 A c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+4 a c c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+2 a c_1 x \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+2 a x+b}{2 (x (a x+b)+c) \left (1+c_1 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )} y(x)\to \frac {2 a x \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+b \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+4 a c+4 A-b^2}{2 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} (x (a x+b)+c)} \end{align*}