20.26.29 problem 23

Internal problem ID [4054]
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 : 23
Date solved : Sunday, March 30, 2025 at 02:15:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x \left (1-x \right ) y^{\prime }+\left (2 x -9\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 48

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+4*x*(1-x)*diff(y(x),x)+(2*x-9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1+\frac {1}{4} x +\frac {1}{20} x^{2}+\frac {1}{120} x^{3}+\frac {1}{840} x^{4}+\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (12+12 x +6 x^{2}+2 x^{3}+\frac {1}{2} x^{4}+\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 90

ode=4*x^2*D[y[x],{x,2}]+4*x*(1-x)*D[y[x],x]+(2*x-9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^{5/2}}{24}+\frac {x^{3/2}}{6}+\frac {1}{x^{3/2}}+\frac {\sqrt {x}}{2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{11/2}}{840}+\frac {x^{9/2}}{120}+\frac {x^{7/2}}{20}+\frac {x^{5/2}}{4}+x^{3/2}\right ) \]

✓ Sympy. Time used: 0.996 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*(1 - x)*Derivative(y(x), x) + (2*x - 9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} \left (\frac {x^{3}}{120} + \frac {x^{2}}{20} + \frac {x}{4} + 1\right ) + \frac {C_{1} \left (\frac {x^{2}}{2} + x + 1\right )}{x^{\frac {3}{2}}} + O\left (x^{6}\right ) \]