Internal problem ID [10400]
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]
\[ \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime }-y^{2}-\left (a x +\mu \right ) y=-\lambda ^{2} x^{2}+\lambda \left (b -\mu \right ) x +c \lambda } \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 476004
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: 23.352 (sec). Leaf size: 433
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 \frac {(x (a x+b)+c)^{\frac {\lambda }{a}-\frac {1}{2}} \exp \left (-\frac {(a (b-2 \mu )+2 b \lambda ) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) \left (\lambda x (x (a x+b)+c)^{\frac {1}{2}-\frac {\lambda }{a}} \exp \left (\frac {(a (b-2 \mu )+2 b \lambda ) \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 (2 b \lambda +a (b-2 \mu )) \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]+x \left (c_1 \lambda (x (a x+b)+c)^{\frac {1}{2}-\frac {\lambda }{a}} \exp \left (\frac {(a (b-2 \mu )+2 b \lambda ) \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right )-a x-b\right )-c\right )}{\int _1^x\exp \left (-\frac {\frac {2 (2 b \lambda +a (b-2 \mu )) \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*}