87.24.18 problem 18

Internal problem ID [23800]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 218
Problem number : 18
Date solved : Thursday, October 02, 2025 at 09:45:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 54

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

\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{8} x^{4}-\frac {1}{8} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}+\frac {1}{2} x^{3}-\frac {13}{24} x^{4}+\frac {8}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {x^5}{8}+\frac {x^4}{8}-\frac {x^3}{6}+1\right )+c_2 \left (\frac {8 x^5}{15}-\frac {13 x^4}{24}+\frac {x^3}{2}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.285 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{8} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (- \frac {13 x^{3}}{24} + \frac {x^{2}}{2} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]