55.2.46 problem 46
Internal
problem
ID
[13272]
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
:
Wednesday, October 01, 2025 at 05:02:19 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational, _Riccati]
\begin{align*} \left (a x +c \right ) y^{\prime }&=\alpha \left (a y+b x \right )^{2}+\beta \left (a y+b x \right )-b x +\gamma \end{align*}
✓ Maple. Time used: 0.016 (sec). Leaf size: 94
ode:=(a*x+c)*diff(y(x),x) = alpha*(b*x+a*y(x))^2+beta*(b*x+a*y(x))-b*x+gamma;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-2 a^{2} \alpha x b -a^{2} \beta +\tan \left (\frac {-2 c_1 \,a^{2}+\ln \left (a x +c \right ) \sqrt {-a^{3} \left (\left (-4 \gamma \alpha +\beta ^{2}\right ) a -4 \alpha b c \right )}}{2 a^{2}}\right ) \sqrt {-a^{3} \left (\left (-4 \gamma \alpha +\beta ^{2}\right ) a -4 \alpha b c \right )}}{2 a^{3} \alpha }
\]
✓ Mathematica. Time used: 60.227 (sec). Leaf size: 98
ode=(a*x+c)*D[y[x],x]==\[Alpha]*(a*y[x]+b*x)^2+\[Beta]*(a*y[x]+b*x)-b*x+\[Gamma];
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 32.723 (sec). Leaf size: 296
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
BETA = symbols("BETA")
Gamma = symbols("Gamma")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-Alpha*(a*y(x) + b*x)**2 - BETA*(a*y(x) + b*x) - Gamma + b*x + (a*x + c)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} - a \sqrt {- \frac {1}{a^{3} \left (4 \mathrm {A} \Gamma a + 4 \mathrm {A} b c - \beta ^{2} a\right )}} \left (- \log {\left (y{\left (x \right )} + \frac {b x}{a} + \frac {- 4 \mathrm {A} \Gamma a^{2} \sqrt {- \frac {1}{a^{3} \left (4 \mathrm {A} \Gamma a + 4 \mathrm {A} b c - \beta ^{2} a\right )}} - 4 \mathrm {A} a b c \sqrt {- \frac {1}{a^{3} \left (4 \mathrm {A} \Gamma a + 4 \mathrm {A} b c - \beta ^{2} a\right )}} + \beta ^{2} a^{2} \sqrt {- \frac {1}{a^{3} \left (4 \mathrm {A} \Gamma a + 4 \mathrm {A} b c - \beta ^{2} a\right )}} + \beta }{2 \mathrm {A} a} \right )} + \log {\left (y{\left (x \right )} + \frac {b x}{a} + \frac {4 \mathrm {A} \Gamma a^{2} \sqrt {- \frac {1}{a^{3} \left (4 \mathrm {A} \Gamma a + 4 \mathrm {A} b c - \beta ^{2} a\right )}} + 4 \mathrm {A} a b c \sqrt {- \frac {1}{a^{3} \left (4 \mathrm {A} \Gamma a + 4 \mathrm {A} b c - \beta ^{2} a\right )}} - \beta ^{2} a^{2} \sqrt {- \frac {1}{a^{3} \left (4 \mathrm {A} \Gamma a + 4 \mathrm {A} b c - \beta ^{2} a\right )}} + \beta }{2 \mathrm {A} a} \right )}\right ) + \frac {\log {\left (a x + c \right )}}{a} = 0
\]