88.24.2 problem 2

Internal problem ID [24195]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 7. Series Methods. Exercises at page 202
Problem number : 2
Date solved : Thursday, October 02, 2025 at 10:00:43 PM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

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

✓ Sympy. Time used: 0.189 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*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)
 

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