18.3.11 problem Problem 16.12 (b)

Internal problem ID [3511]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 16, Series solutions of ODEs. Section 16.6 Exercises, page 550
Problem number : Problem 16.12 (b)
Date solved : Sunday, March 30, 2025 at 01:45:24 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (z^{2}+5 z +7\right ) y^{\prime \prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

Order:=6; 
ode:=(z^2+5*z+7)*diff(diff(y(z),z),z)+2*y(z) = 0; 
dsolve(ode,y(z),type='series',z=0);
 

\[ y = \left (1-\frac {1}{7} z^{2}+\frac {5}{147} z^{3}-\frac {11}{2058} z^{4}+\frac {5}{14406} z^{5}\right ) y \left (0\right )+\left (z -\frac {1}{21} z^{3}+\frac {5}{294} z^{4}-\frac {47}{10290} z^{5}\right ) y^{\prime }\left (0\right )+O\left (z^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

ode=(z^2+5*z+7)*D[y[z],{z,2}]+2*y[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,5}]
 

\[ y(z)\to c_2 \left (-\frac {47 z^5}{10290}+\frac {5 z^4}{294}-\frac {z^3}{21}+z\right )+c_1 \left (\frac {5 z^5}{14406}-\frac {11 z^4}{2058}+\frac {5 z^3}{147}-\frac {z^2}{7}+1\right ) \]

✓ Sympy. Time used: 0.798 (sec). Leaf size: 44

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq((z**2 + 5*z + 7)*Derivative(y(z), (z, 2)) + 2*y(z),0) 
ics = {} 
dsolve(ode,func=y(z),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (z \right )} = C_{2} \left (- \frac {11 z^{4}}{2058} + \frac {5 z^{3}}{147} - \frac {z^{2}}{7} + 1\right ) + C_{1} z \left (\frac {5 z^{3}}{294} - \frac {z^{2}}{21} + 1\right ) + O\left (z^{6}\right ) \]