30.13.3 problem 3

Internal problem ID [7635]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 04:55:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{80} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{8} x^{4}+\frac {1}{16} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (\frac {x^5}{80}+\frac {x^4}{48}+\frac {x^3}{12}+1\right )+c_2 \left (\frac {x^5}{16}+\frac {x^4}{8}+\frac {x^3}{6}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 1.071 (sec). Leaf size: 80

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} \sin ^{2}{\left (x \right )}}{96} + \frac {x^{4} \sin {\left (x \right )}}{24} + \frac {x^{3} \sin {\left (x \right )}}{12} + \frac {x^{2} \sin {\left (x \right )}}{4} + 1\right ) + C_{1} x \left (\frac {x^{3} \sin {\left (x \right )}}{24} + \frac {x^{3}}{12} + \frac {x^{2} \sin {\left (x \right )}}{12} + \frac {x^{2}}{6} + \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]