73.23.17 problem 33.5 (e)

Internal problem ID [15592]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.5 (e)
Date solved : Monday, March 31, 2025 at 01:41:54 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-x^{2}+4\right ) y^{\prime \prime }-5 x y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 39

Order:=6; 
ode:=(-x^2+4)*diff(diff(y(x),x),x)-5*x*diff(y(x),x)-3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {3}{8} x^{2}+\frac {15}{128} x^{4}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{10} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

ode=(4-x^2)*D[y[x],{x,2}]-5*x*D[y[x],x]-3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{10}+\frac {x^3}{3}+x\right )+c_1 \left (\frac {15 x^4}{128}+\frac {3 x^2}{8}+1\right ) \]

✓ Sympy. Time used: 0.876 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*x*Derivative(y(x), x) + (4 - x**2)*Derivative(y(x), (x, 2)) - 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {15 x^{4}}{128} + \frac {3 x^{2}}{8} + 1\right ) + C_{1} x \left (\frac {x^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]