67.26.15 problem 36.6 (c)

Internal problem ID [17042]
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 : 36.6 (c)
Date solved : Thursday, October 02, 2025 at 01:42:15 PM
CAS classification : [_Laguerre]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.027 (sec). Leaf size: 52

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

\[ y = c_1 \,x^{2} \left (1-\frac {2}{3} x +\frac {1}{12} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\ln \left (x \right ) \left (12 x^{2}-8 x^{3}+x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (-2-8 x -7 x^{2}+\frac {58}{3} x^{3}-\frac {25}{6} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 70

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

\[ y(x)\to c_2 \left (\frac {x^4}{12}-\frac {2 x^3}{3}+x^2\right )+c_1 \left (\frac {1}{6} \left (14 x^4-70 x^3+39 x^2+24 x+6\right )-\frac {1}{2} x^2 \left (x^2-8 x+12\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.319 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (\frac {x^{2}}{12} - \frac {2 x}{3} + 1\right ) + O\left (x^{6}\right ) \]