61.31.29 problem 210
Internal
problem
ID
[12631]
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
:
210
Date
solved
:
Monday, March 31, 2025 at 06:48:40 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (1-k \right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} y&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 63
ode:=2*x*(a*x^2+b*x+c)*diff(diff(y(x),x),x)+(a*(2-k)*x^2+b*(1-k)*x-c*k)*diff(y(x),x)+lambda*x^(k+1)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (c_1 \,{\mathrm e}^{i \sqrt {2}\, \int \sqrt {\frac {\lambda \,x^{k}}{a \,x^{2}+b x +c}}d x}+c_2 \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \int \sqrt {\frac {\lambda \,x^{k}}{a \,x^{2}+b x +c}}d x}{2}}
\]
✓ Mathematica. Time used: 117.122 (sec). Leaf size: 774
ode=2*x*(a*x^2+b*x+c)*D[y[x],{x,2}]+(a*(2-k)*x^2+b*(1-k)*x-c*k)*D[y[x],x]+(\[Lambda]*x^(k+1))*y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt {2} \sqrt {c_1} \tan \left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}-c_2\right )}{\sqrt {-\sec ^2\left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}-c_2\right )}} \\
y(x)\to -\frac {\sqrt {2} \sqrt {c_1} \tan \left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}+c_2\right )}{\sqrt {-\sec ^2\left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}+c_2\right )}} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
k = symbols("k")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(lambda_*x**(k + 1)*y(x) + 2*x*(a*x**2 + b*x + c)*Derivative(y(x), (x, 2)) + (a*x**2*(2 - k) + b*x*(1 - k) - c*k)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None