61.31.13 problem 194
Internal
problem
ID
[12615]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-6
Problem
number
:
194
Date
solved
:
Monday, March 31, 2025 at 05:54:07 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} \left (x +a_{2} \right ) y^{\prime \prime }+x \left (b_{1} x +a_{1} \right ) y^{\prime }+\left (b_{0} x +a_{0} \right ) y&=0 \end{align*}
✓ Maple. Time used: 0.113 (sec). Leaf size: 319
ode:=x^2*(x+a__2)*diff(diff(y(x),x),x)+x*(b__1*x+a__1)*diff(y(x),x)+(b__0*x+a__0)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 \,x^{\frac {a_{2} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -b_{1} a_{2} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}, \frac {b_{1} a_{2} -a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}\right ], \left [\frac {a_{2} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )+c_2 \,x^{-\frac {-a_{2} +a_{1} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{2 a_{2}}} \operatorname {hypergeom}\left (\left [-\frac {\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2} -b_{1} a_{2} +\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+a_{1}}{2 a_{2}}, \frac {b_{1} a_{2} -a_{1} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}+\sqrt {b_{1}^{2}-4 b_{0} -2 b_{1} +1}\, a_{2}}{2 a_{2}}\right ], \left [\frac {a_{2} -\sqrt {a_{2}^{2}+\left (-4 a_{0} -2 a_{1} \right ) a_{2} +a_{1}^{2}}}{a_{2}}\right ], -\frac {x}{a_{2}}\right )
\]
✓ Mathematica. Time used: 0.722 (sec). Leaf size: 384
ode=x^2*(x+a2)*D[y[x],{x,2}]+x*(b1*x+a1)*D[y[x],x]+(b0*x+a0)*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \text {a2}^{-\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}-\text {a1}+\text {a2}}{2 \text {a2}}} x^{-\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}+\text {a1}-\text {a2}}{2 \text {a2}}} \left (c_2 x^{\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}}{\text {a2}}} \operatorname {Hypergeometric2F1}\left (\frac {-\text {a1}+\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}-\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},\frac {-\text {a1}+\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}+\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2}^2-4 \text {a0} \text {a2}}}{\text {a2}},-\frac {x}{\text {a2}}\right )+c_1 \text {a2}^{\frac {\sqrt {-4 \text {a0} \text {a2}+\text {a1}^2-2 \text {a1} \text {a2}+\text {a2}^2}}{\text {a2}}} \operatorname {Hypergeometric2F1}\left (-\frac {\text {a1}-\text {a2} \text {b1}+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}+\text {a2} \sqrt {(\text {b1}-1)^2-4 \text {b0}}}{2 \text {a2}},-\frac {\text {a1}-\text {a2} \left (\text {b1}+\sqrt {(\text {b1}-1)^2-4 \text {b0}}\right )+\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2} (\text {a2}-4 \text {a0})}}{2 \text {a2}},1-\frac {\sqrt {\text {a1}^2-2 \text {a2} \text {a1}+\text {a2}^2-4 \text {a0} \text {a2}}}{\text {a2}},-\frac {x}{\text {a2}}\right )\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
a__0 = symbols("a__0")
a__1 = symbols("a__1")
a__2 = symbols("a__2")
b__0 = symbols("b__0")
b__1 = symbols("b__1")
y = Function("y")
ode = Eq(x**2*(a__2 + x)*Derivative(y(x), (x, 2)) + x*(a__1 + b__1*x)*Derivative(y(x), x) + (a__0 + b__0*x)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None