84.30.8 problem 18.18

Internal problem ID [22297]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 18. Linear differential equations with variable coefficients. Supplementary problems
Problem number : 18.18
Date solved : Thursday, October 02, 2025 at 08:37:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} -1 \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 306

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

\[ y(x)\to c_1 e^{-\frac {2 \sqrt {2}}{\sqrt {x+1}}} \left (\frac {41522222767994171 (x+1)^5}{3647915698170101760}+\frac {1504867749296087 (x+1)^{9/2}}{79798155897470976 \sqrt {2}}-\frac {3810475992017 (x+1)^4}{184717953466368}-\frac {242570969281 (x+1)^{7/2}}{2164663517184 \sqrt {2}}+\frac {1176536189 (x+1)^3}{19327352832}+\frac {12980205 (x+1)^{5/2}}{33554432 \sqrt {2}}-\frac {284063 (x+1)^2}{6291456}-\frac {46613 (x+1)^{3/2}}{49152 \sqrt {2}}-\frac {303 (x+1)}{1024}+\frac {19 \sqrt {x+1}}{16 \sqrt {2}}+1\right ) (x+1)^{7/4}+c_2 e^{\frac {2 \sqrt {2}}{\sqrt {x+1}}} \left (\frac {41522222767994171 (x+1)^5}{3647915698170101760}-\frac {1504867749296087 (x+1)^{9/2}}{79798155897470976 \sqrt {2}}-\frac {3810475992017 (x+1)^4}{184717953466368}+\frac {242570969281 (x+1)^{7/2}}{2164663517184 \sqrt {2}}+\frac {1176536189 (x+1)^3}{19327352832}-\frac {12980205 (x+1)^{5/2}}{33554432 \sqrt {2}}-\frac {284063 (x+1)^2}{6291456}+\frac {46613 (x+1)^{3/2}}{49152 \sqrt {2}}-\frac {303 (x+1)}{1024}-\frac {19 \sqrt {x+1}}{16 \sqrt {2}}+1\right ) (x+1)^{7/4} \]

✗ Sympy

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

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