82.3.3 problem 25-3
Internal
problem
ID
[21786]
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
:
25-3
Date
solved
:
Thursday, October 02, 2025 at 08:02:10 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 256
Order:=6;
ode:=diff(diff(y(x),x),x)+p(x)*diff(y(x),x)+q(x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (1-\frac {q \left (0\right ) x^{2}}{2}+\left (\frac {p \left (0\right ) q \left (0\right )}{6}-\frac {D\left (q \right )\left (0\right )}{6}\right ) x^{3}+\left (-\frac {p \left (0\right )^{2} q \left (0\right )}{24}+\frac {q \left (0\right )^{2}}{24}+\frac {D\left (p \right )\left (0\right ) q \left (0\right )}{12}+\frac {p \left (0\right ) D\left (q \right )\left (0\right )}{24}-\frac {D^{\left (2\right )}\left (q \right )\left (0\right )}{24}\right ) x^{4}+\left (\frac {p \left (0\right )^{3} q \left (0\right )}{120}-\frac {D\left (q \right )\left (0\right ) p \left (0\right )^{2}}{120}+\frac {\left (-5 D\left (p \right )\left (0\right ) q \left (0\right )-2 q \left (0\right )^{2}+D^{\left (2\right )}\left (q \right )\left (0\right )\right ) p \left (0\right )}{120}+\frac {\left (4 D\left (q \right )\left (0\right )+3 D^{\left (2\right )}\left (p \right )\left (0\right )\right ) q \left (0\right )}{120}+\frac {D\left (p \right )\left (0\right ) D\left (q \right )\left (0\right )}{40}-\frac {D^{\left (3\right )}\left (q \right )\left (0\right )}{120}\right ) x^{5}\right ) y \left (0\right )+\left (x -\frac {p \left (0\right ) x^{2}}{2}+\frac {\left (p \left (0\right )^{2}-q \left (0\right )-D\left (p \right )\left (0\right )\right ) x^{3}}{6}+\frac {\left (-p \left (0\right )^{3}+\left (2 q \left (0\right )+3 D\left (p \right )\left (0\right )\right ) p \left (0\right )-2 D\left (q \right )\left (0\right )-D^{\left (2\right )}\left (p \right )\left (0\right )\right ) x^{4}}{24}+\frac {\left (p \left (0\right )^{4}+\left (-3 q \left (0\right )-6 D\left (p \right )\left (0\right )\right ) p \left (0\right )^{2}+\left (5 D\left (q \right )\left (0\right )+4 D^{\left (2\right )}\left (p \right )\left (0\right )\right ) p \left (0\right )+q \left (0\right )^{2}+4 D\left (p \right )\left (0\right ) q \left (0\right )+3 D\left (p \right )\left (0\right )^{2}-3 D^{\left (2\right )}\left (q \right )\left (0\right )-D^{\left (3\right )}\left (p \right )\left (0\right )\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right )
\]
✓ Mathematica. Time used: 0.001 (sec). Leaf size: 57
ode=D[y[x],{x,2}]+p(x)*D[y[x],x]+q(x)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\[
y(x)\to c_2 \left (\frac {p^2 x^5}{40}-\frac {p x^3}{6}-\frac {q x^4}{12}+x\right )+c_1 \left (\frac {1}{40} p q x^5-\frac {q x^3}{6}+1\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
p = Function("p")
q = Function("q")
ode = Eq(p(x)*Derivative(y(x), x) + q(x)*y(x) + Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
ValueError : ODE p(x)*Derivative(y(x), x) + q(x)*y(x) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular