67.25.17 problem 35.4 (c)

Internal problem ID [17012]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (c)
Date solved : Thursday, October 02, 2025 at 01:41:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 58

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

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {4}{5} x +\frac {4}{15} x^{2}-\frac {16}{315} x^{3}+\frac {2}{315} x^{4}-\frac {8}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (256 x^{4}-\frac {1024}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-192 x -192 x^{2}-256 x^{3}+\frac {6144}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 79

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

\[ y(x)\to c_1 \left (\frac {4 x^4+16 x^3+12 x^2+12 x+9}{9 x^2}-\frac {16}{9} x^2 \log (x)\right )+c_2 \left (\frac {2 x^6}{315}-\frac {16 x^5}{315}+\frac {4 x^4}{15}-\frac {4 x^3}{5}+x^2\right ) \]

✓ Sympy. Time used: 0.269 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (- \frac {16 x^{3}}{315} + \frac {4 x^{2}}{15} - \frac {4 x}{5} + 1\right ) + O\left (x^{6}\right ) \]