28.4.4 problem 4

Internal problem ID [7193]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 4
Date solved : Tuesday, September 30, 2025 at 04:25:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{3} y^{\prime \prime }-\left (2 x -1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✗ Maple

Order:=6; 
ode:=x^3*diff(diff(y(x),x),x)-(2*x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 222

ode=x^3*D[y[x],{x,2}]-(2*x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 e^{-\frac {2 i}{\sqrt {x}}} x^{3/4} \left (-\frac {1159525191825 i x^{9/2}}{8796093022208}+\frac {218243025 i x^{7/2}}{4294967296}-\frac {405405 i x^{5/2}}{8388608}+\frac {3465 i x^{3/2}}{8192}+\frac {75369137468625 x^5}{281474976710656}-\frac {41247931725 x^4}{549755813888}+\frac {11486475 x^3}{268435456}-\frac {45045 x^2}{524288}-\frac {945 x}{512}-\frac {35 i \sqrt {x}}{16}+1\right )+c_2 e^{\frac {2 i}{\sqrt {x}}} x^{3/4} \left (\frac {1159525191825 i x^{9/2}}{8796093022208}-\frac {218243025 i x^{7/2}}{4294967296}+\frac {405405 i x^{5/2}}{8388608}-\frac {3465 i x^{3/2}}{8192}+\frac {75369137468625 x^5}{281474976710656}-\frac {41247931725 x^4}{549755813888}+\frac {11486475 x^3}{268435456}-\frac {45045 x^2}{524288}-\frac {945 x}{512}+\frac {35 i \sqrt {x}}{16}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*Derivative(y(x), (x, 2)) - (2*x - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**3*Derivative(y(x), (x, 2)) - (2*x - 1)*y(x) does not match hint 2nd_power_series_regular