67.25.22 problem 35.4 (h)

Internal problem ID [17017]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (h)
Date solved : Thursday, October 02, 2025 at 01:41:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+\left (-2 x^{3}+5 x \right ) y^{\prime }+\left (-x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 31

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)+(-2*x^3+5*x)*diff(y(x),x)+(-x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 34

ode=2*x^2*D[y[x],{x,2}]+(5*x-2*x^3)*D[y[x],x]+(1-x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.376 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + (1 - x**2)*y(x) + (-2*x**3 + 5*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2} \left (- \frac {x^{6}}{528} - \frac {x^{4}}{56} - \frac {x^{2}}{6} + 1\right )}{x} + \frac {C_{1}}{\sqrt {x}} + O\left (x^{6}\right ) \]