22.5.15 problem 9.15

Internal problem ID [4602]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.15
Date solved : Tuesday, September 30, 2025 at 07:36:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 43

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

\[ y = c_1 x \left (1+\frac {2}{3} x +\frac {1}{4} x^{2}+\frac {1}{15} x^{3}+\frac {1}{72} x^{4}+\frac {1}{420} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2+x^{2}+\frac {2}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {x^3}{8}-\frac {x^2}{3}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^5}{72}+\frac {x^4}{15}+\frac {x^3}{4}+\frac {2 x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.310 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{72} + \frac {x^{3}}{15} + \frac {x^{2}}{4} + \frac {2 x}{3} + 1\right ) + \frac {C_{1}}{x} + O\left (x^{6}\right ) \]