20.26.10 problem 2

Internal problem ID [4035]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 2
Date solved : Sunday, March 30, 2025 at 02:15:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+2 x^{2} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 46

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

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x +\frac {3}{64} x^{2}-\frac {5}{768} x^{3}+\frac {35}{49152} x^{4}-\frac {21}{327680} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{64} x^{2}+\frac {1}{256} x^{3}-\frac {19}{32768} x^{4}+\frac {25}{393216} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 129

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {21 x^5}{327680}+\frac {35 x^4}{49152}-\frac {5 x^3}{768}+\frac {3 x^2}{64}-\frac {x}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {25 x^5}{393216}-\frac {19 x^4}{32768}+\frac {x^3}{256}-\frac {x^2}{64}\right )+\sqrt {x} \left (-\frac {21 x^5}{327680}+\frac {35 x^4}{49152}-\frac {5 x^3}{768}+\frac {3 x^2}{64}-\frac {x}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.836 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {35 x^{4}}{49152} - \frac {5 x^{3}}{768} + \frac {3 x^{2}}{64} - \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]