35.9.19 problem 10, using series method

Internal problem ID [6254]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 12, Series Solutions of Differential Equations. Section 1. Miscellaneous problems. page 564
Problem number : 10, using series method
Date solved : Sunday, March 30, 2025 at 10:44:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 34

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

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

✓ Sympy. Time used: 0.812 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*Derivative(y(x), x) + (4*x**2 - 2)*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)
 

\[ y{\left (x \right )} = \frac {7 x^{5} r{\left (3 \right )}}{10} + C_{2} \left (\frac {x^{4}}{2} + x^{2} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{5}\right ) + O\left (x^{6}\right ) \]