61.2.46 problem 46

Internal problem ID [12052]
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
Date solved : Monday, January 27, 2025 at 11:54:51 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Riccati]

\begin{align*} \left (a x +c \right ) y^{\prime }&=\alpha \left (b x +a y\right )^{2}+\beta \left (b x +a y\right )-b x +\gamma \end{align*}

✓ Solution by Maple

Time used: 0.010 (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)
 

\[ y = \frac {-2 a^{2} x \alpha b -a^{2} \beta +\sqrt {-a^{3} \left (\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 \alpha b c \right )}\, \tan \left (\frac {-2 c_{1} a^{2}+\ln \left (a x +c \right ) \sqrt {-a^{3} \left (\left (-4 \alpha \gamma +\beta ^{2}\right ) a -4 \alpha b c \right )}}{2 a^{2}}\right )}{2 a^{3} \alpha } \]

✓ Solution by Mathematica

Time used: 60.340 (sec). Leaf size: 98

DSolve[(a*x+c)*D[y[x],x]==\[Alpha]*(a*y[x]+b*x)^2+\[Beta]*(a*y[x]+b*x)-b*x+\[Gamma],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\frac {-a \alpha \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}} \tan \left (\frac {1}{2} a \alpha \log (a x+c) \sqrt {\frac {4 a \alpha \gamma -a \beta ^2+4 \alpha b c}{a^3 \alpha ^2}}+c_1\right )+2 \alpha b x+\beta }{2 a \alpha } \]