40.18.1 problem 20

Internal problem ID [6813]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 27. The Legendre, Bessel and Gauss Equations. Supplemetary problems. Page 230
Problem number : 20
Date solved : Sunday, March 30, 2025 at 11:23:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} z^{\prime \prime }+t z^{\prime }+\left (t^{2}-\frac {1}{9}\right ) z&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 39

Order:=6; 
ode:=diff(diff(z(t),t),t)+t*diff(z(t),t)+(t^2-1/9)*z(t) = 0; 
dsolve(ode,z(t),type='series',t=0);
 

\[ z = \left (1+\frac {1}{18} t^{2}-\frac {179}{1944} t^{4}\right ) z \left (0\right )+\left (t -\frac {4}{27} t^{3}-\frac {139}{4860} t^{5}\right ) z^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

ode=D[z[t],{t,2}]+t*D[z[t],t]+(t^2-1/9)*z[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},z[t],{t,0,5}]
 

\[ z(t)\to c_2 \left (-\frac {139 t^5}{4860}-\frac {4 t^3}{27}+t\right )+c_1 \left (-\frac {179 t^4}{1944}+\frac {t^2}{18}+1\right ) \]

✓ Sympy. Time used: 0.984 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
z = Function("z") 
ode = Eq(t*Derivative(z(t), t) + (t**2 - 1/9)*z(t) + Derivative(z(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=z(t),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ z{\left (t \right )} = - \frac {13 t^{5} r{\left (3 \right )}}{90} + C_{2} \left (- \frac {179 t^{4}}{1944} + \frac {t^{2}}{18} + 1\right ) + C_{1} t \left (1 - \frac {t^{4}}{20}\right ) + O\left (t^{6}\right ) \]