18.3.7 problem Problem 16.9

Internal problem ID [3507]
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 16.9
Date solved : Sunday, March 30, 2025 at 01:45:18 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }-2 z y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 37

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

\[ y = \left (1+z^{2}+\frac {1}{2} z^{4}\right ) y \left (0\right )+\left (z +\frac {2}{3} z^{3}+\frac {4}{15} z^{5}\right ) y^{\prime }\left (0\right )+O\left (z^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 38

ode=D[y[z],{z,2}]-2*z*D[y[z],z]-2*y[z]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[z],{z,0,5}]
 

\[ y(z)\to c_2 \left (\frac {4 z^5}{15}+\frac {2 z^3}{3}+z\right )+c_1 \left (\frac {z^4}{2}+z^2+1\right ) \]

✓ Sympy. Time used: 0.759 (sec). Leaf size: 29

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

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