18.3.10 problem Problem 16.12 (a)

Internal problem ID [3510]
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 (a)
Date solved : Tuesday, March 04, 2025 at 04:43:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (z^{2}+5 z +6\right ) y^{\prime \prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 54

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

\[ y \left (z \right ) = \left (1-\frac {1}{6} z^{2}+\frac {5}{108} z^{3}-\frac {13}{1296} z^{4}+\frac {5}{2592} z^{5}\right ) y \left (0\right )+\left (z -\frac {1}{18} z^{3}+\frac {5}{216} z^{4}-\frac {17}{2160} z^{5}\right ) D\left (y \right )\left (0\right )+O\left (z^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 63

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

\[ y(z)\to c_2 \left (-\frac {17 z^5}{2160}+\frac {5 z^4}{216}-\frac {z^3}{18}+z\right )+c_1 \left (\frac {5 z^5}{2592}-\frac {13 z^4}{1296}+\frac {5 z^3}{108}-\frac {z^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.923 (sec). Leaf size: 44

from sympy import * 
z = symbols("z") 
y = Function("y") 
ode = Eq((z**2 + 5*z + 6)*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)
 

\[ y{\left (z \right )} = C_{2} \left (- \frac {13 z^{4}}{1296} + \frac {5 z^{3}}{108} - \frac {z^{2}}{6} + 1\right ) + C_{1} z \left (\frac {5 z^{3}}{216} - \frac {z^{2}}{18} + 1\right ) + O\left (z^{6}\right ) \]