82.3.35 problem 25-35
Internal
problem
ID
[21818]
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
:
25-35
Date
solved
:
Thursday, October 02, 2025 at 08:02:34 PM
CAS
classification
:
[_Laguerre]
\begin{align*} x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+p y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 154
Order:=6;
ode:=x*diff(diff(y(x),x),x)+(1-x)*diff(y(x),x)+p*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (\left (2 p +1\right ) x +\left (\frac {1}{4} p +\frac {1}{4}-\frac {3}{4} p^{2}\right ) x^{2}+\left (-\frac {2}{9} p^{2}+\frac {1}{27} p +\frac {1}{18}+\frac {11}{108} p^{3}\right ) x^{3}+\left (\frac {7}{192} p^{3}-\frac {167}{3456} p^{2}+\frac {1}{192} p +\frac {1}{96}-\frac {25}{3456} p^{4}\right ) x^{4}+\left (\frac {719}{86400} p^{3}-\frac {61}{21600} p^{4}+\frac {137}{432000} p^{5}+\frac {1}{600}-\frac {37}{4320} p^{2}+\frac {1}{1500} p \right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-p x +\frac {1}{4} \left (-1+p \right ) p x^{2}-\frac {1}{36} \left (p -2\right ) \left (-1+p \right ) p x^{3}+\frac {1}{576} \left (p -3\right ) \left (p -2\right ) \left (-1+p \right ) p x^{4}-\frac {1}{14400} \left (p -4\right ) \left (p -3\right ) \left (p -2\right ) \left (-1+p \right ) p x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 415
ode=x*D[y[x],{x,2}]+(1-x)*D[y[x],x]+p*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\[
y(x)\to c_1 \left (-\frac {(p-4) (p-3) (p-2) (p-1) p x^5}{14400}+\frac {1}{576} (p-3) (p-2) (p-1) p x^4-\frac {1}{36} (p-2) (p-1) p x^3+\frac {1}{4} (p-1) p x^2-p x+1\right )+c_2 \left (\frac {(p-4) (p-3) (p-2) (p-1) x^5}{14400}+\frac {(p-4) (p-3) (p-2) p x^5}{14400}+\frac {(p-4) (p-3) (p-1) p x^5}{14400}+\frac {(p-4) (p-2) (p-1) p x^5}{14400}+\frac {137 (p-4) (p-3) (p-2) (p-1) p x^5}{432000}+\frac {(p-3) (p-2) (p-1) p x^5}{14400}-\frac {1}{576} (p-3) (p-2) (p-1) x^4-\frac {1}{576} (p-3) (p-2) p x^4-\frac {1}{576} (p-3) (p-1) p x^4-\frac {25 (p-3) (p-2) (p-1) p x^4}{3456}-\frac {1}{576} (p-2) (p-1) p x^4+\frac {1}{36} (p-2) (p-1) x^3+\frac {1}{36} (p-2) p x^3+\frac {11}{108} (p-2) (p-1) p x^3+\frac {1}{36} (p-1) p x^3-\frac {1}{4} (p-1) x^2-\frac {3}{4} (p-1) p x^2-\frac {p x^2}{4}+\left (-\frac {(p-4) (p-3) (p-2) (p-1) p x^5}{14400}+\frac {1}{576} (p-3) (p-2) (p-1) p x^4-\frac {1}{36} (p-2) (p-1) p x^3+\frac {1}{4} (p-1) p x^2-p x+1\right ) \log (x)+2 p x+x\right )
\]
✓ Sympy. Time used: 0.347 (sec). Leaf size: 73
from sympy import *
x = symbols("x")
p = symbols("p")
y = Function("y")
ode = Eq(p*y(x) + x*Derivative(y(x), (x, 2)) + (1 - x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
\[
y{\left (x \right )} = C_{1} \left (- \frac {p x^{5} \left (p - 4\right ) \left (p - 3\right ) \left (p - 2\right ) \left (p - 1\right )}{14400} + \frac {p x^{4} \left (p - 3\right ) \left (p - 2\right ) \left (p - 1\right )}{576} - \frac {p x^{3} \left (p - 2\right ) \left (p - 1\right )}{36} + \frac {p x^{2} \left (p - 1\right )}{4} - p x + 1\right ) + O\left (x^{6}\right )
\]