14.25.19 problem 20

Internal problem ID [4024]
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 11.4. page 758
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 07:00:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 38

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

\[ y = \sqrt {x}\, \left (\left (x +\frac {1}{4} x^{2}+\frac {1}{18} x^{3}+\frac {1}{96} x^{4}+\frac {1}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 60

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

\[ y(x)\to c_2 \left (\sqrt {x} \left (\frac {x^5}{600}+\frac {x^4}{96}+\frac {x^3}{18}+\frac {x^2}{4}+x\right )+\sqrt {x} \log (x)\right )+c_1 \sqrt {x} \]

✓ Sympy. Time used: 0.300 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = C_{1} \sqrt {x} + O\left (x^{6}\right ) \]