46.1.18 problem 16

Internal problem ID [9521]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 06:20:01 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 35

Order:=8; 
ode:=(x^2+1)*diff(diff(y(x),x),x)-6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 32

ode=(x^2+1)*D[y[x],{x,2}]-6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

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

✓ Sympy. Time used: 0.226 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 1)*Derivative(y(x), (x, 2)) - 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

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