56.4.50 problem 47

Internal problem ID [8939]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 47
Date solved : Sunday, March 30, 2025 at 01:55:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 34

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-1/4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 58

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-1/4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^{7/2}}{24}-\frac {x^{3/2}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{9/2}}{120}-\frac {x^{5/2}}{6}+\sqrt {x}\right ) \]

✓ Sympy. Time used: 0.928 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 1/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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