Internal problem ID [9646]
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]
Solve \begin {gather*} \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 =0} \end {gather*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 5761
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)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 14.486 (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]
\begin{align*} 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 } \text {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 ) \\ \end{align*}