56.4.55 problem 52

Internal problem ID [8944]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 52
Date solved : Wednesday, March 05, 2025 at 07:09:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }-\left (x^{2}+\frac {5}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)-(x^2+5/4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} x^{3} \left (1+\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12-6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 58

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

\[ y(x)\to c_1 \left (-\frac {x^{7/2}}{8}-\frac {x^{3/2}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{13/2}}{280}+\frac {x^{9/2}}{10}+x^{5/2}\right ) \]

✓ Sympy. Time used: 0.878 (sec). Leaf size: 37

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 + 5/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {5}{2}} \left (\frac {x^{2}}{10} + 1\right ) + \frac {C_{1} \left (- \frac {x^{4}}{8} - \frac {x^{2}}{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]