50.4.52 problem 49

Internal problem ID [10227]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 49
Date solved : Tuesday, September 30, 2025 at 07:12:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{2}+6 x \right ) y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 44

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

\[ y = c_1 \left (1-\frac {1}{6} x +\frac {1}{42} x^{2}-\frac {1}{336} x^{3}+\frac {1}{3024} x^{4}-\frac {1}{30240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (2880-2880 x +1440 x^{2}-480 x^{3}+120 x^{4}-24 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{5}} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 68

ode=x^2*D[y[x],{x,2}]+(6*x+x^2)*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^4}{3024}-\frac {x^3}{336}+\frac {x^2}{42}-\frac {x}{6}+1\right )+c_1 \left (\frac {1}{x^5}-\frac {1}{x^4}+\frac {1}{2 x^3}-\frac {1}{6 x^2}+\frac {1}{24 x}\right ) \]

✓ Sympy. Time used: 0.338 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{5}}{30240} + \frac {x^{4}}{3024} - \frac {x^{3}}{336} + \frac {x^{2}}{42} - \frac {x}{6} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right )}{x^{5}} + O\left (x^{6}\right ) \]