7.12.12 problem 12(a)

Internal problem ID [2424]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8, Series solutions. Page 195
Problem number : 12(a)
Date solved : Tuesday, September 30, 2025 at 05:35:40 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} y^{\prime \prime }+t^{3} y^{\prime }+3 t^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 29

Order:=6; 
ode:=diff(diff(y(t),t),t)+t^3*diff(y(t),t)+3*t^2*y(t) = 0; 
dsolve(ode,y(t),type='series',t=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

ode=D[y[t],{t,2}]+t^3*D[y[t],t]+3*t^2*y[t]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[t],{t,0,5}]
 

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

✓ Sympy. Time used: 0.215 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**3*Derivative(y(t), t) + 3*t**2*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)
 

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