33.6.6 problem Problem 27.39

Internal problem ID [7851]
Book : Schaums Outline Differential Equations, 4th edition. Bronson and Costa. McGraw Hill 2014
Section : Chapter 27. Power series solutions of linear DE with variable coefficients. Supplementary Problems. page 274
Problem number : Problem 27.39
Date solved : Tuesday, September 30, 2025 at 05:06:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-x^{2} y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 49

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

\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (\frac {x^5}{120}+\frac {x^4}{12}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^5}{20}+\frac {x^4}{24}+\frac {x^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.210 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{24} + \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (\frac {x^{3}}{12} + \frac {x^{2}}{6} + 1\right ) + O\left (x^{6}\right ) \]