14.26.4 problem (a)

Internal problem ID [4029]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : (a)
Date solved : Tuesday, September 30, 2025 at 07:00:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 46

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

\[ y = x \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{2} x^{2}-\frac {5}{18} x^{3}+\frac {19}{144} x^{4}-\frac {167}{3600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {41}{108} x^{3}-\frac {89}{432} x^{4}+\frac {2281}{27000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 114

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

\[ y(x)\to c_1 x \left (-\frac {167 x^5}{3600}+\frac {19 x^4}{144}-\frac {5 x^3}{18}+\frac {x^2}{2}-x+1\right )+c_2 \left (x \left (\frac {2281 x^5}{27000}-\frac {89 x^4}{432}+\frac {41 x^3}{108}-\frac {3 x^2}{4}+x\right )+x \left (-\frac {167 x^5}{3600}+\frac {19 x^4}{144}-\frac {5 x^3}{18}+\frac {x^2}{2}-x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.316 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} x \left (\frac {19 x^{4}}{144} - \frac {5 x^{3}}{18} + \frac {x^{2}}{2} - x + 1\right ) + O\left (x^{6}\right ) \]