61.2.57 problem 57
Internal
problem
ID
[11984]
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
:
57
Date
solved
:
Wednesday, March 05, 2025 at 03:29:38 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\gamma &=0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 858
ode:=(a*x^2+b)*diff(y(x),x)+alpha*y(x)^2+beta*x*y(x)+gamma = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.636 (sec). Leaf size: 598
ode=(a*x^2+b)*D[y[x],x]+\[Alpha]*y[x]^2+\[Beta]*x*y[x]+\[Gamma]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {i \left (c_1 \left (\sqrt {4 a \alpha \gamma +b \beta ^2}-2 a \sqrt {b}-\sqrt {b} \beta \right ) P_{\frac {\beta }{2 a}+1}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+2 i \sqrt {a} x (a+\beta ) Q_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+\left (\sqrt {4 a \alpha \gamma +b \beta ^2}-2 a \sqrt {b}-\sqrt {b} \beta \right ) Q_{\frac {\beta }{2 a}+1}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right )-2 \sqrt {a} c_1 x (a+\beta ) P_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{2 \sqrt {a} \alpha \left (c_1 P_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )+Q_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )\right )} \\
y(x)\to \frac {-2 x (a+\beta )+\frac {i \left (\sqrt {4 a \alpha \gamma +b \beta ^2}-2 a \sqrt {b}-\sqrt {b} \beta \right ) P_{\frac {\beta }{2 a}+1}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} P_{\frac {\beta }{2 a}}^{\frac {\sqrt {b \beta ^2+4 a \alpha \gamma }}{2 a \sqrt {b}}}\left (\frac {i \sqrt {a} x}{\sqrt {b}}\right )}}{2 \alpha } \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
BETA = symbols("BETA")
Gamma = symbols("Gamma")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(Alpha*y(x)**2 + BETA*x*y(x) + Gamma + (a*x**2 + b)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out