Internal problem ID [10389]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power
Functions
Problem number: 59.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\[ \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime }-y^{2}-\left (2 \lambda x +b \right ) y=\lambda \left (\lambda -a \right ) x^{2}+\mu } \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 542
dsolve((a*x^2+b*x+c)*diff(y(x),x)=y(x)^2+(2*lambda*x+b)*y(x)+lambda*(lambda-a)*x^2+mu,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {8 \left (\left (i a \sqrt {4 a c -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}-\sqrt {-4 a c +b^{2}}\, \left (2 x \lambda +b \right )\right ) c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-\left (i a \sqrt {4 a c -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}+\sqrt {-4 a c +b^{2}}\, \left (2 x \lambda +b \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ) \left (a \,x^{2}+b x +c \right )^{2} a^{2}}{\sqrt {-4 a c +b^{2}}\, \left (2 a x -\sqrt {-4 a c +b^{2}}+b \right ) \left (2 a x +\sqrt {-4 a c +b^{2}}+b \right ) \left (i \sqrt {4 a c -b^{2}}+2 a x +b \right ) \left (-b +i \sqrt {4 a c -b^{2}}-2 a x \right ) \left (c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )} \]
✓ Solution by Mathematica
Time used: 17.168 (sec). Leaf size: 93
DSolve[(a*x^2+b*x+c)*y'[x]==y[x]^2+(2*\[Lambda]*x+b)*y[x]+\[Lambda]*(\[Lambda]-a)*x^2+\[Mu],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{2} \left (\sqrt {4 (c \lambda +\mu )-b^2} \tan \left (\frac {\sqrt {-b^2+4 c \lambda +4 \mu } \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+c_1\right )-b-2 \lambda x\right ) \]