Internal problem ID [9633]
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: 46.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational, _Riccati]
Solve \begin {gather*} \boxed {\left (x a +c \right ) y^{\prime }-\alpha \left (a y+x b \right )^{2}-\beta \left (a y+x b \right )+x b -\gamma =0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 94
dsolve((a*x+c)*diff(y(x),x)=alpha*(a*y(x)+b*x)^2+beta*(a*y(x)+b*x)-b*x+gamma,y(x), singsol=all)
\[ y \relax (x ) = \frac {-2 a^{2} \alpha b x -a^{2} \beta +\tan \left (\frac {-2 c_{1} a^{2}+\ln \left (a x +c \right ) \sqrt {a^{3} \left (4 \alpha \gamma a -a \,\beta ^{2}+4 \alpha b c \right )}}{2 a^{2}}\right ) \sqrt {a^{3} \left (4 \alpha \gamma a -a \,\beta ^{2}+4 \alpha b c \right )}}{2 a^{3} \alpha } \]
✓ Solution by Mathematica
Time used: 60.333 (sec). Leaf size: 98
DSolve[(a*x+c)*y'[x]==\[Alpha]*(a*y[x]+b*x)^2+\[Beta]*(a*y[x]+b*x)-b*x+\[Gamma],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {-a \alpha \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}} \tan \left (\frac {1}{2} a \alpha \log (a x+c) \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}+c_1\right )+2 \alpha b x+\beta }{2 a \alpha } \\ \end{align*}