82.3.20 problem 25-20
Internal
problem
ID
[21803]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
II.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
25.
Power
series
about
a
singular
point.
Page
762
Problem
number
:
25-20
Date
solved
:
Thursday, October 02, 2025 at 08:02:24 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 \left (1-x \right ) y^{\prime \prime }+\left (1+x \right ) y^{\prime }+\left (x -3-\left (x -1\right )^{2} {\mathrm e}^{x}\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 1 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 74
Order:=6;
ode:=2*(1-x)*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)+(x-3-(x-1)^2*exp(x))*y(x) = 0;
dsolve(ode,y(x),type='series',x=1);
\[
y = c_1 \left (x -1\right )^{2} \left (1+\frac {1}{16} \left (x -1\right )^{2}+\left (-\frac {{\mathrm e}}{30}+\frac {1}{240}\right ) \left (x -1\right )^{3}+\left (-\frac {11 \,{\mathrm e}}{480}+\frac {1}{640}\right ) \left (x -1\right )^{4}+\left (-\frac {11 \,{\mathrm e}}{1120}+\frac {1}{6720}\right ) \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+c_2 \left (-2-2 \left (x -1\right )-\frac {3}{4} \left (x -1\right )^{2}+\left (-\frac {1}{3}+\frac {{\mathrm e}}{3}\right ) \left (x -1\right )^{3}+\left (-\frac {13}{192}+\frac {13 \,{\mathrm e}}{48}\right ) \left (x -1\right )^{4}+\left (-\frac {1}{64}+\frac {37 \,{\mathrm e}}{240}\right ) \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )
\]
✓ Mathematica. Time used: 0.029 (sec). Leaf size: 90
ode=2*(1-x)*D[y[x],{x,2}]+(1+x)*D[y[x],x]+(x-3-(x-1)^2*Exp[x])*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
\[
y(x)\to c_1 \left (-\frac {13}{384} (4 e-1) (x-1)^4-\frac {1}{6} (e-1) (x-1)^3+\frac {3}{8} (x-1)^2+x\right )+c_2 \left (-\frac {(44 e-3) (x-1)^6}{1920}-\frac {1}{240} (8 e-1) (x-1)^5+\frac {1}{16} (x-1)^4+(x-1)^2\right )
\]
✓ Sympy. Time used: 1.038 (sec). Leaf size: 116
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((2 - 2*x)*Derivative(y(x), (x, 2)) + (x + 1)*Derivative(y(x), x) + (x - (x - 1)**2*exp(x) - 3)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=1,n=6)
\[
y{\left (x \right )} = C_{1} \left (x - 1\right )^{2} \left (\frac {\left (x - 1\right )^{3} \left (- 8 x^{2} \left (\frac {3 x e^{x}}{2} - 1\right ) - 3 \left (x^{2} + \left (x e^{x} - 1\right ) \left (\frac {3 x e^{x}}{2} - 1\right )\right ) \left (\frac {3 x e^{x}}{2} - 2\right ) + 24 e^{x}\right )}{720} + \frac {\left (x - 1\right )^{2} \left (x^{2} + \left (x e^{x} - 1\right ) \left (\frac {3 x e^{x}}{2} - 1\right )\right )}{16} - \frac {\left (x - 1\right ) \left (\frac {3 x e^{x}}{2} - 1\right )}{3} + 1\right ) + O\left (x^{6}\right )
\]