46.5.6 problem 16

Internal problem ID [7342]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. REVIEW QUESTIONS. page 201
Problem number : 16
Date solved : Sunday, March 30, 2025 at 11:54:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 35

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

\[ y = c_1 \,x^{2} \left (1-\frac {1}{2} x^{2}+\frac {9}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-6 x^{2}+\frac {9}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (\frac {3 x^3}{8}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {9 x^6}{56}-\frac {x^4}{2}+x^2\right ) \]

✓ Sympy. Time used: 1.067 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {x^{2}}{2} - 1\right ) + \frac {C_{1} \left (- \frac {7 x^{6}}{48} + \frac {3 x^{4}}{8} - \frac {x^{2}}{2} + 1\right )}{x} + O\left (x^{6}\right ) \]