Internal problem ID [9647]
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: 60.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {\left (a \,x^{2}+x b +c \right ) y^{\prime }-y^{2}-\left (x a +\mu \right ) y+\lambda ^{2} x^{2}-\lambda \left (b -\mu \right ) x -c \lambda =0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 470863
dsolve((a*x^2+b*x+c)*diff(y(x),x)=y(x)^2+(a*x+mu)*y(x)-lambda^2*x^2+lambda*(b-mu)*x+lambda*c,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 11.561 (sec). Leaf size: 183
DSolve[(a*x^2+b*x+c)*y'[x]==y[x]^2+(a*x+\[Mu])*y[x]-\[Lambda]^2*x^2+\[Lambda]*(b-\[Mu])*x+\[Lambda]*c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \lambda x-\frac {(x (a x+b)+c)^{\frac {\lambda }{a}+\frac {1}{2}} \exp \left (-\frac {(a (b-2 \mu )+2 b \lambda ) \text {ArcTan}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right )}{\int _1^x\exp \left (-\frac {\frac {2 (a b+2 \lambda b-2 a \mu ) \text {ArcTan}\left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+(a-2 \lambda ) \log (c+K[1] (b+a K[1]))}{2 a}\right )dK[1]+c_1} \\ y(x)\to \lambda x \\ \end{align*}