20.26.32 problem 26

Internal problem ID [4057]
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 : 26
Date solved : Tuesday, March 04, 2025 at 05:23:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x \left (1+2 x \right ) y^{\prime }+\left (4 x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 46

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+4*x*(2*x+1)*diff(y(x),x)+(4*x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{1} x \left (1-x +\frac {2}{3} x^{2}-\frac {1}{3} x^{3}+\frac {2}{15} x^{4}-\frac {2}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 88

ode=4*x^2*D[y[x],{x,2}]+4*x*(1+2*x)*D[y[x],x]+(4*x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {2 x^{7/2}}{3}-\frac {4 x^{5/2}}{3}+2 x^{3/2}-2 \sqrt {x}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {2 x^{9/2}}{15}-\frac {x^{7/2}}{3}+\frac {2 x^{5/2}}{3}-x^{3/2}+\sqrt {x}\right ) \]

✓ Sympy. Time used: 0.859 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*(2*x + 1)*Derivative(y(x), x) + (4*x - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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