18.3.4 problem Problem 16.4

Internal problem ID [3504]
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 16.4
Date solved : Tuesday, March 04, 2025 at 04:43:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} f^{\prime \prime }+2 \left (z -1\right ) f^{\prime }+4 f&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 57

Order:=6; 
ode:=diff(diff(f(z),z),z)+2*(z-1)*diff(f(z),z)+4*f(z) = 0; 
dsolve(ode,f(z),type='series',z=0);
 

\[ f \left (z \right ) = \left (1-2 z^{2}-\frac {4}{3} z^{3}+\frac {2}{3} z^{4}+\frac {14}{15} z^{5}\right ) f \left (0\right )+\left (z +z^{2}-\frac {1}{3} z^{3}-\frac {5}{6} z^{4}-\frac {1}{6} z^{5}\right ) D\left (f \right )\left (0\right )+O\left (z^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 127

ode=D[ f[z],{z,2}]+2*(z-a)*D[ f[z],z]+4*f[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},f[z],{z,0,5}]
 

\[ f(z)\to c_1 \left (-\frac {4}{15} a^3 z^5-\frac {2 a^2 z^4}{3}+\frac {6 a z^5}{5}-\frac {4 a z^3}{3}+\frac {4 z^4}{3}-2 z^2+1\right )+c_2 \left (\frac {2 a^4 z^5}{15}+\frac {a^3 z^4}{3}-\frac {4 a^2 z^5}{5}+\frac {2 a^2 z^3}{3}-\frac {7 a z^4}{6}+a z^2+\frac {z^5}{2}-z^3+z\right ) \]

✓ Sympy. Time used: 0.820 (sec). Leaf size: 46

from sympy import * 
z = symbols("z") 
f = Function("f") 
ode = Eq((2*z - 2)*Derivative(f(z), z) + 4*f(z) + Derivative(f(z), (z, 2)),0) 
ics = {} 
dsolve(ode,func=f(z),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ f{\left (z \right )} = C_{2} \left (\frac {2 z^{4}}{3} - \frac {4 z^{3}}{3} - 2 z^{2} + 1\right ) + C_{1} z \left (- \frac {5 z^{3}}{6} - \frac {z^{2}}{3} + z + 1\right ) + O\left (z^{6}\right ) \]