Internal problem ID [3633]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 13
Problem number: 377.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {\left (c \,x^{2}+b x +a \right )^{2} \left (y^{\prime }+y^{2}\right )=-A} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 846
dsolve((c*x^2+b*x+a)^2*(diff(y(x),x)+y(x)^2)+A = 0,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2 \left (-i \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}\, \sqrt {4 a c -b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} c +i \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}\, \sqrt {4 a c -b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c +2 \sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} c x +2 \sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c x +\sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1} b +\sqrt {-4 a c +b^{2}}\, {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} b \right ) c}{\sqrt {-4 a c +b^{2}}\, \left (2 c x +b +i \sqrt {4 a c -b^{2}}\right ) \left (i \sqrt {4 a c -b^{2}}-2 c x -b \right ) \left (c_{1} {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {c \sqrt {-\frac {4 a c -b^{2}+4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )} \]
✓ Solution by Mathematica
Time used: 3.457 (sec). Leaf size: 743
DSolve[(a+b x+c x^2)^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 {b+2 c x}{\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 {b+2 c x}{\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 {b+2 c x}{\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 {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+2 c 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 {b+2 c x}{\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}}+b+2 c x}{2 (a+x (b+c x)) \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 {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )} y(x)\to \frac {2 c 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}} (a+x (b+c x))} \end{align*}