45.2.2 problem 2

Internal problem ID [7225]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page 239
Problem number : 2
Date solved : Sunday, March 30, 2025 at 11:51:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x +3\right )^{2} y^{\prime \prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 58

Order:=6; 
ode:=x*(x+3)^2*diff(diff(y(x),x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1+\frac {1}{18} x -\frac {11}{972} x^{2}+\frac {277}{104976} x^{3}-\frac {12539}{18895680} x^{4}+\frac {893821}{5101833600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (\frac {1}{9} x +\frac {1}{162} x^{2}-\frac {11}{8748} x^{3}+\frac {277}{944784} x^{4}-\frac {12539}{170061120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {5}{108} x^{2}+\frac {167}{26244} x^{3}-\frac {13583}{11337408} x^{4}+\frac {1327279}{5101833600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 87

ode=x*(x+3)^2*D[y[x],{x,2}]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x \left (277 x^3-1188 x^2+5832 x+104976\right ) \log (x)}{944784}+\frac {3037 x^4+864 x^3-174960 x^2+6298560 x+11337408}{11337408}\right )+c_2 \left (-\frac {12539 x^5}{18895680}+\frac {277 x^4}{104976}-\frac {11 x^3}{972}+\frac {x^2}{18}+x\right ) \]

✓ Sympy. Time used: 1.014 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 3)**2*Derivative(y(x), (x, 2)) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{2880} + \frac {x^{3}}{144} + \frac {x^{2}}{12} + \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]