67.26.8 problem 36.2 (h)

Internal problem ID [17035]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (h)
Date solved : Thursday, October 02, 2025 at 01:42:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 62

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

\[ y = \frac {c_1 \,x^{2} \left (1+\frac {2}{3} x +\frac {1}{6} x^{2}+\frac {1}{45} x^{3}+\frac {1}{540} x^{4}+\frac {1}{9450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (4 x^{2}+\frac {8}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+4 x -\frac {32}{9} x^{3}-\frac {25}{18} x^{4}-\frac {157}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \left (\frac {31 x^4+88 x^3+36 x^2-72 x+36}{36 x}-\frac {1}{3} x \left (x^2+4 x+6\right ) \log (x)\right )+c_2 \left (\frac {x^5}{540}+\frac {x^4}{45}+\frac {x^3}{6}+\frac {2 x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.262 (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) - (2*x + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{540} + \frac {x^{3}}{45} + \frac {x^{2}}{6} + \frac {2 x}{3} + 1\right ) + O\left (x^{6}\right ) \]