56.4.29 problem 25

Internal problem ID [8918]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 25
Date solved : Sunday, March 30, 2025 at 01:53:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (11 x^{2}+11 x +9\right ) y^{\prime }+\left (7 x^{2}+10 x +6\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 45

Order:=6; 
ode:=2*x^2*(x^2+x+1)*diff(diff(y(x),x),x)+x*(11*x^2+11*x+9)*diff(y(x),x)+(7*x^2+10*x+6)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {1}{8} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}}+\frac {c_2 \left (1-\frac {1}{3} x +\frac {2}{5} x^{2}-\frac {5}{21} x^{3}+\frac {7}{135} x^{4}+\frac {76}{1155} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{2}}} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 83

ode=2*x^2*(1+x+x^2)*D[y[x],{x,2}] + x*(9+11*x+11*x^2)*D[y[x],x] + (6+10*x+7*x^2)*y[x] == 0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_2 \left (\frac {x^5}{30}+\frac {x^4}{8}-\frac {x^3}{3}+\frac {x^2}{2}+1\right )}{x^2}+\frac {c_1 \left (\frac {76 x^5}{1155}+\frac {7 x^4}{135}-\frac {5 x^3}{21}+\frac {2 x^2}{5}-\frac {x}{3}+1\right )}{x^{3/2}} \]

✓ Sympy. Time used: 1.315 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x**2 + x + 1)*Derivative(y(x), (x, 2)) + x*(11*x**2 + 11*x + 9)*Derivative(y(x), x) + (7*x**2 + 10*x + 6)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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