56.4.48 problem 45

Internal problem ID [8937]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 45
Date solved : Sunday, March 30, 2025 at 01:55:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+4 x 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: 0.025 (sec). Leaf size: 34

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+(x^2+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x +c_2 \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 40

ode=x^2*D[y[x],{x,2}]+4*x*D[y[x],x]+(x^2+2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^3}{120}-\frac {x}{6}+\frac {1}{x}\right )+c_1 \left (\frac {x^2}{24}+\frac {1}{x^2}-\frac {1}{2}\right ) \]

✓ Sympy. Time used: 0.849 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) + (x**2 + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (- \frac {x^{6}}{5040} + \frac {x^{4}}{120} - \frac {x^{2}}{6} + 1\right )}{x} + O\left (1\right ) \]