88.25.1 problem 2 (a)

Internal problem ID [24206]
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 212
Problem number : 2 (a)
Date solved : Thursday, October 02, 2025 at 10:00:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 36

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

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 71

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

\[ y(x)\to \frac {c_1 \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )}{x}+c_2 \left (\frac {\frac {x^2}{4}-\frac {3 x^4}{128}}{x}+\frac {\left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 0.265 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = \frac {C_{1} \left (- \frac {x^{6}}{2304} + \frac {x^{4}}{64} - \frac {x^{2}}{4} + 1\right )}{x} + O\left (x^{6}\right ) \]