87.26.14 problem 23

Internal problem ID [23848]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 253
Problem number : 23
Date solved : Thursday, October 02, 2025 at 09:45:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{16}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 35

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-1/16)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{3} x^{2}+\frac {1}{42} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{4}}}+c_2 \,x^{{1}/{4}} \left (1-\frac {1}{5} x^{2}+\frac {1}{90} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 52

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-1/16)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (\frac {x^4}{90}-\frac {x^2}{5}+1\right )+\frac {c_2 \left (\frac {x^4}{42}-\frac {x^2}{3}+1\right )}{\sqrt [4]{x}} \]

✓ Sympy. Time used: 0.341 (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 - 1/16)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [4]{x} \left (\frac {x^{4}}{90} - \frac {x^{2}}{5} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{42} - \frac {x^{2}}{3} + 1\right )}{\sqrt [4]{x}} + O\left (x^{6}\right ) \]