58.14.2 problem 2

Internal problem ID [14842]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.1. Exercises page 232
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:55:41 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+8 x y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

Order:=6; 
ode:=diff(diff(y(x),x),x)+8*x*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 38

ode=D[y[x],{x,2}]+8*x*D[y[x],x]-4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.187 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*x*Derivative(y(x), x) - 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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