7.12.1 problem 1

Internal problem ID [2413]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 05:35:33 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+t y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 39

Order:=6; 
ode:=diff(diff(y(t),t),t)+t*diff(y(t),t)+y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

\[ y = \left (1-\frac {1}{2} t^{2}+\frac {1}{8} t^{4}\right ) y \left (0\right )+\left (t -\frac {1}{3} t^{3}+\frac {1}{15} t^{5}\right ) y^{\prime }\left (0\right )+O\left (t^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

\[ y(t)\to c_2 \left (\frac {t^5}{15}-\frac {t^3}{3}+t\right )+c_1 \left (\frac {t^4}{8}-\frac {t^2}{2}+1\right ) \]

✓ Sympy. Time used: 0.190 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = C_{2} \left (\frac {t^{4}}{8} - \frac {t^{2}}{2} + 1\right ) + C_{1} t \left (1 - \frac {t^{2}}{3}\right ) + O\left (t^{6}\right ) \]