82.2.17 problem 24-18
Internal
problem
ID
[21778]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
II.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
24.
Power
series
about
an
ordinary
point.
Page
719
Problem
number
:
24-18
Date
solved
:
Thursday, October 02, 2025 at 08:02:02 PM
CAS
classification
:
[_Jacobi]
\begin{align*} -a b y+\left (c -\left (1+a +b \right ) x \right ) y^{\prime }+x \left (1-x \right ) y^{\prime \prime }&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 447
Order:=6;
ode:=x*(1-x)*diff(diff(y(x),x),x)+(c-(a+b+1)*x)*diff(y(x),x)-a*b*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_1 \,x^{-c +1} \left (1-\frac {\left (b -c +1\right ) \left (a -c +1\right )}{c -2} x +\frac {1}{2} \frac {\left (b -c +2\right ) \left (b -c +1\right ) \left (a -c +2\right ) \left (a -c +1\right )}{\left (c -2\right ) \left (c -3\right )} x^{2}-\frac {1}{6} \frac {\left (-c +3+b \right ) \left (b -c +2\right ) \left (b -c +1\right ) \left (-c +3+a \right ) \left (a -c +2\right ) \left (a -c +1\right )}{\left (c -2\right ) \left (c -3\right ) \left (c -4\right )} x^{3}+\frac {1}{24} \frac {\left (-c +4+b \right ) \left (-c +3+b \right ) \left (b -c +2\right ) \left (b -c +1\right ) \left (a -c +1\right ) \left (-c +4+a \right ) \left (-c +3+a \right ) \left (a -c +2\right )}{\left (c -2\right ) \left (c -3\right ) \left (c -4\right ) \left (c -5\right )} x^{4}-\frac {1}{120} \frac {\left (-c +4+b \right ) \left (-c +3+b \right ) \left (b -c +2\right ) \left (b -c +1\right ) \left (-c +5+b \right ) \left (a -c +1\right ) \left (-c +5+a \right ) \left (-c +4+a \right ) \left (-c +3+a \right ) \left (a -c +2\right )}{\left (c -2\right ) \left (c -3\right ) \left (c -4\right ) \left (c -5\right ) \left (c -6\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {a b}{c} x +\frac {1}{2} \frac {a b \left (b +1\right ) \left (a +1\right )}{c \left (c +1\right )} x^{2}+\frac {1}{6} \frac {a b \left (b +2\right ) \left (b +1\right ) \left (a +2\right ) \left (a +1\right )}{c \left (c +1\right ) \left (c +2\right )} x^{3}+\frac {1}{24} \frac {a b \left (b +3\right ) \left (b +2\right ) \left (b +1\right ) \left (a +3\right ) \left (a +2\right ) \left (a +1\right )}{c \left (c +1\right ) \left (c +2\right ) \left (c +3\right )} x^{4}+\frac {1}{120} \frac {a b \left (b +4\right ) \left (b +3\right ) \left (b +2\right ) \left (b +1\right ) \left (a +4\right ) \left (a +3\right ) \left (a +2\right ) \left (a +1\right )}{c \left (c +1\right ) \left (c +2\right ) \left (c +3\right ) \left (c +4\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )
\]
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 2568
ode=x*(1-x)*D[y[x],{x,2}]+(c-(a+b+1)*x)*D[y[x],x]-a*b*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\begin{align*} \text {Solution too large to show}\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*y(x) + x*(1 - x)*Derivative(y(x), (x, 2)) + (c - x*(a + b + 1))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
ValueError : Expected Expr or iterable but got None