88.25.4 problem 2 (d)

Internal problem ID [24209]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 212
Problem number : 2 (d)
Date solved : Thursday, October 02, 2025 at 10:00:50 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime \prime }+\frac {x^{2} y^{\prime \prime }}{1+x}-\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=x^4*diff(diff(diff(y(x),x),x),x)+x^2/(1+x)*diff(diff(y(x),x),x)-(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.207 (sec). Leaf size: 322

ode=x^4*D[y[x],{x,3}]+x^2/(1+x)*D[y[x],{x,2}]-(1+x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 e^{\frac {1}{x}} \left (\frac {815704 x^5}{15}+\frac {63545 x^4}{12}+\frac {1687 x^3}{3}+65 x^2+8 x+1\right ) x^5+c_3 \left (\frac {\left (-5701301-2549699 \sqrt {5}\right ) x^5}{320 \left (49795+22269 \sqrt {5}\right )}+\frac {\left (16155+7223 \sqrt {5}\right ) x^4}{73632+32928 \sqrt {5}}+\frac {\left (-67-35 \sqrt {5}\right ) x^3}{48 \left (9+4 \sqrt {5}\right )}+\frac {3 x^2}{2 \left (7+3 \sqrt {5}\right )}+\frac {\left (7-\sqrt {5}\right ) x}{2 \left (1+\sqrt {5}\right )}+1\right ) x^{\frac {1}{2} \left (1+\sqrt {5}\right )}+c_2 \left (\frac {\left (5701301-2549699 \sqrt {5}\right ) x^5}{320 \left (22269 \sqrt {5}-49795\right )}+\frac {\left (7223 \sqrt {5}-16155\right ) x^4}{32928 \sqrt {5}-73632}+\frac {\left (67-35 \sqrt {5}\right ) x^3}{48 \left (4 \sqrt {5}-9\right )}+\frac {3 x^2}{2 \left (7-3 \sqrt {5}\right )}+\frac {\left (-7-\sqrt {5}\right ) x}{2 \left (\sqrt {5}-1\right )}+1\right ) x^{\frac {1}{2} \left (1-\sqrt {5}\right )} \]

✗ Sympy

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

Series solution not supported for ode of order > 2