52.2.18 problem 18

Internal problem ID [8269]
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. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 18
Date solved : Sunday, March 30, 2025 at 12:50:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }-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: 0.032 (sec). Leaf size: 37

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{6} x^{2}+\frac {1}{168} x^{4}-\frac {1}{11088} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_2 x \left (1-\frac {1}{10} x^{2}+\frac {1}{360} x^{4}-\frac {1}{28080} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 62

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

\[ y(x)\to c_1 x \left (-\frac {x^6}{28080}+\frac {x^4}{360}-\frac {x^2}{10}+1\right )+c_2 \sqrt {x} \left (-\frac {x^6}{11088}+\frac {x^4}{168}-\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.943 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) - 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=8)
 

\[ y{\left (x \right )} = C_{2} x \left (- \frac {x^{6}}{28080} + \frac {x^{4}}{360} - \frac {x^{2}}{10} + 1\right ) + C_{1} \sqrt {x} \left (- \frac {x^{6}}{11088} + \frac {x^{4}}{168} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{8}\right ) \]