23.3.512 problem 518
Internal
problem
ID
[6226]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
3.
THE
DIFFERENTIAL
EQUATION
IS
LINEAR
AND
OF
SECOND
ORDER,
page
311
Problem
number
:
518
Date
solved
:
Friday, October 03, 2025 at 01:57:04 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (\operatorname {b2} x +\operatorname {a2} \right ) y+x \left (\operatorname {b1} x +\operatorname {a1} \right ) y^{\prime }+\left (1-x \right ) x^{2} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 240
ode:=(b2*x+a2)*y(x)+x*(b1*x+a1)*diff(y(x),x)+(1-x)*x^2*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = x^{-\frac {\operatorname {a1}}{2}} \sqrt {x}\, \left (c_1 \,x^{\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}} \operatorname {hypergeom}\left (\left [-\frac {\operatorname {b1}}{2}-\frac {\operatorname {a1}}{2}+\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}+\frac {\sqrt {\operatorname {b1}^{2}+2 \operatorname {b1} +4 \operatorname {b2} +1}}{2}, -\frac {\operatorname {b1}}{2}-\frac {\operatorname {a1}}{2}+\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}-\frac {\sqrt {\operatorname {b1}^{2}+2 \operatorname {b1} +4 \operatorname {b2} +1}}{2}\right ], \left [1+\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}\right ], x\right )+c_2 \,x^{-\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}} \operatorname {hypergeom}\left (\left [-\frac {\operatorname {b1}}{2}-\frac {\operatorname {a1}}{2}-\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}+\frac {\sqrt {\operatorname {b1}^{2}+2 \operatorname {b1} +4 \operatorname {b2} +1}}{2}, -\frac {\operatorname {b1}}{2}-\frac {\operatorname {a1}}{2}-\frac {\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}}{2}-\frac {\sqrt {\operatorname {b1}^{2}+2 \operatorname {b1} +4 \operatorname {b2} +1}}{2}\right ], \left [1-\sqrt {\operatorname {a1}^{2}-2 \operatorname {a1} -4 \operatorname {a2} +1}\right ], x\right )\right )
\]
✓ Mathematica. Time used: 0.299 (sec). Leaf size: 317
ode=(a2 + b2*x)*y[x] + x*(a1 + b1*x)*D[y[x],x] + (1 - x)*x^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to i^{-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1} x^{\frac {1}{2} \left (-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\text {a1}+1\right )} \left (c_2 i^{2 \sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}} x^{\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-\text {a1}-\text {b1}+\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right ),\frac {1}{2} \left (-\text {a1}-\text {b1}+\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right ),\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+1,x\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (-\text {a1}-\text {b1}-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}-\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right ),\frac {1}{2} \left (-\text {a1}-\text {b1}-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1}+\sqrt {\text {b1}^2+2 \text {b1}+4 \text {b2}+1}\right ),1-\sqrt {\text {a1}^2-2 \text {a1}-4 \text {a2}+1},x\right )\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a1 = symbols("a1")
a2 = symbols("a2")
b1 = symbols("b1")
b2 = symbols("b2")
y = Function("y")
ode = Eq(x**2*(1 - x)*Derivative(y(x), (x, 2)) + x*(a1 + b1*x)*Derivative(y(x), x) + (a2 + b2*x)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ValueError : Expected Expr or iterable but got None