61.2.59 problem 59
Internal
problem
ID
[11986]
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
:
59
Date
solved
:
Wednesday, March 05, 2025 at 03:38:08 PM
CAS
classification
:
[_rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]
\begin{align*} \left (a \,x^{2}+b x +c \right ) y^{\prime }&=y^{2}+\left (2 \lambda x +b \right ) y+\lambda \left (\lambda -a \right ) x^{2}+\mu \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 530
ode:=(a*x^2+b*x+c)*diff(y(x),x) = y(x)^2+(2*lambda*x+b)*y(x)+lambda*(lambda-a)*x^2+mu;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {8 a^{2} \left (c_{1} \left (i a \sqrt {4 a c -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}-\sqrt {-4 a c +b^{2}}\, \left (2 \lambda x +b \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-\left (i a \sqrt {4 a c -b^{2}}\, \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}+\sqrt {-4 a c +b^{2}}\, \left (2 \lambda x +b \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ) \left (a \,x^{2}+b x +c \right )^{2}}{\sqrt {-4 a c +b^{2}}\, \left (2 a x +b -\sqrt {-4 a c +b^{2}}\right ) \left (2 a x +b +\sqrt {-4 a c +b^{2}}\right ) \left (i \sqrt {4 a c -b^{2}}+2 a x +b \right ) \left (-b +i \sqrt {4 a c -b^{2}}-2 a x \right ) \left (c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {b^{2}-4 c \lambda -4 \mu }{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )}
\]
✓ Mathematica. Time used: 1.414 (sec). Leaf size: 80
ode=(a*x^2+b*x+c)*D[y[x],x]==y[x]^2+(2*\[Lambda]*x+b)*y[x]+\[Lambda]*(\[Lambda]-a)*x^2+\[Mu];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {1}{2} \left (\sqrt {4 (c \lambda +\mu )-b^2} \tan \left (\frac {1}{2} \sqrt {-b^2+4 c \lambda +4 \mu } \int _1^x\frac {1}{c+K[5] (b+a K[5])}dK[5]+c_1\right )-b-2 \lambda x\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
cg = symbols("cg")
mu = symbols("mu")
y = Function("y")
ode = Eq(-cg*x**2*(-a + cg) - mu - (b + 2*cg*x)*y(x) + (a*x**2 + b*x + c)*Derivative(y(x), x) - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out