30.13.5 problem 5

Internal problem ID [7637]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 04:55:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 59

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

\[ x = \left (1+\frac {1}{2} t^{2}-\frac {1}{12} t^{3}+\frac {13}{96} t^{4}-\frac {1}{16} t^{5}\right ) x \left (0\right )+\left (t +\frac {1}{4} t^{2}+\frac {1}{4} t^{3}-\frac {1}{96} t^{4}+\frac {31}{480} t^{5}\right ) x^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 70

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

\[ x(t)\to c_1 \left (-\frac {t^5}{16}+\frac {13 t^4}{96}-\frac {t^3}{12}+\frac {t^2}{2}+1\right )+c_2 \left (\frac {31 t^5}{480}-\frac {t^4}{96}+\frac {t^3}{4}+\frac {t^2}{4}+t\right ) \]

✓ Sympy. Time used: 0.438 (sec). Leaf size: 44

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

\[ x{\left (t \right )} = C_{2} \left (\frac {13 t^{4}}{96} - \frac {t^{3}}{12} + \frac {t^{2}}{2} + 1\right ) + C_{1} t \left (- \frac {t^{3}}{96} + \frac {t^{2}}{4} + \frac {t}{4} + 1\right ) + O\left (t^{6}\right ) \]