29.10.3 problem 269
Internal
problem
ID
[4869]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
10
Problem
number
:
269
Date
solved
:
Tuesday, March 04, 2025 at 07:24:48 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x^{2} y^{\prime }&=a +b x y+c \,x^{4} y^{2} \end{align*}
✓ Maple. Time used: 0.000 (sec). Leaf size: 111
ode:=x^2*diff(y(x),x) = a+b*x*y(x)+c*x^4*y(x)^2;
dsolve(ode,y(x), singsol=all);
\[
y \left (x \right ) = -\frac {a \left (\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right )\right )}{x \left (x \sqrt {a c}\, \left (\operatorname {BesselY}\left (\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right )\right )+\left (b +1\right ) \left (\operatorname {BesselY}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right ) c_{1} +\operatorname {BesselJ}\left (-\frac {1}{2}-\frac {b}{2}, x \sqrt {a c}\right )\right )\right )}
\]
✓ Mathematica. Time used: 0.329 (sec). Leaf size: 394
ode=x^2 D[y[x],x]==a+b x y[x]+c x^4 y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {\sqrt {a} \sqrt {c} x \operatorname {BesselY}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+(b+3) \operatorname {BesselY}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} x \operatorname {BesselY}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )+\sqrt {a} \sqrt {c} c_1 x \operatorname {BesselJ}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+b c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )+3 c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} c_1 x \operatorname {BesselJ}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )}{2 c x^3 \left (\operatorname {BesselY}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )+c_1 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )\right )} \\
y(x)\to -\frac {\sqrt {a} \sqrt {c} x \operatorname {BesselJ}\left (\frac {b+1}{2},\sqrt {a} \sqrt {c} x\right )+(b+3) \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )-\sqrt {a} \sqrt {c} x \operatorname {BesselJ}\left (\frac {b+5}{2},\sqrt {a} \sqrt {c} x\right )}{2 c x^3 \operatorname {BesselJ}\left (\frac {b+3}{2},\sqrt {a} \sqrt {c} x\right )} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-a - b*x*y(x) - c*x**4*y(x)**2 + x**2*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out